Springer Verlag Numerical Methods for Elliptic and Parabolic Partial Differential Equations eBook.KB

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NumericalMethodsforEllipticandParabolicPartialDifferentialEquationsPeterKnabnerLutzAngermannSpringer TextsinAppliedMathematics44EditorsJ.E.MarsdenL.SirovichS.S.AntmanAdvisorsG.IoossP.HolmesD.BarkleyM.DellnitzP.Newton Thispageintentionallyleftblank PeterKnabnerLutzAngermannNumericalMethodsforEllipticandParabolicPartialDifferentialEquationsWith67Figures PeterKnabnerLutzAngermannInstituteforAppliedMathematicsInstituteforMathematicsUniversityofErlangenUniversityofClausthalMartensstrasse3Erzstrasse1D-91058ErlangenD-38678Clausthal-ZellerfeldGermanyGermanyknabner@am.uni-erlangen.deangermann@math.tu-clausthal.deSeriesEditorsJ.E.MarsdenL.SirovichControlandDynamicalSystems,107–81DivisionofAppliedMathematicsCaliforniaInstituteofTechnologyBrownUniversityPasadena,CA91125Providence,RI02912USAUSAmarsden@cds.caltech.educhico@camelot.mssm.eduS.S.AntmanDepartmentofMathematicsandInstituteforPhysicalScienceandTechnologyUniversityofMarylandCollegePark,MD20742-4015USAssa@math.umd.eduMathematicsSubjectClassification(2000):65Nxx,65Mxx,65F10,65H10LibraryofCongressCataloging-in-PublicationDataKnabner,Peter.[NumerikpartiellerDifferentialgleichungen.English]Numericalmethodsforellipticandparabolicpartialdifferentialequations/PeterKnabner,LutzAngermann.p.cm.—(Textsinappliedmathematics;44)Includebibliographicalreferencesandindex.ISBN0-387-95449-X(alk.paper)1.Differentialequations,Partial—Numericalsolutions.I.Angermann,Lutz.II.Title.III.Series.QA377.K5752003515′.353—dc212002044522ISBN0-387-95449-XPrintedonacid-freepaper.2003Springer-VerlagNewYork,Inc.Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.PrintedintheUnitedStatesofAmerica.987654321SPIN10867187Typesetting:Pagescreatedbytheauthorsin2eusingSpringer’ssvsing6.clsmacro.www.springer-ny.comSpringer-VerlagNewYorkBerlinHeidelbergAmemberofBertelsmannSpringerScience+BusinessMediaGmbH SeriesPrefaceMathematicsisplayinganevermoreimportantroleinthephysicalandbiologicalsciences,provokingablurringofboundariesbetweenscientificdisciplinesandaresurgenceofinterestinthemodernaswellastheclassicaltechniquesofappliedmathematics.Thisrenewalofinterest,bothinre-searchandteaching,hasledtotheestablishmentoftheseriesTextsinAppliedMathematics(TAM).Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevelofexcitementontheresearchfrontierasnewertechniques,suchasnumeri-calandsymboliccomputersystems,dynamicalsystems,andchaos,mixwithandreinforcethetraditionalmethodsofappliedmathematics.Thus,thepurposeofthistextbookseriesistomeetthecurrentandfutureneedsoftheseadvancesandtoencouragetheteachingofnewcourses.TAMwillpublishtextbookssuitableforuseinadvancedundergraduateandbeginninggraduatecourses,andwillcomplementtheAppliedMathe-maticalSciences(AMS)series,whichwillfocusonadvancedtextbooksandresearch-levelmonographs.Pasadena,CaliforniaJ.E.MarsdenProvidence,RhodeIslandL.SirovichCollegePark,MarylandS.S.Antman 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PrefacetotheEnglishEditionShortlyaftertheappearanceoftheGermaneditionwewereaskedbySpringertocreateanEnglishversionofourbook,andwegratefullyac-cepted.Wetookthisopportunitynotonlytocorrectsomemisprintsandmistakesthathavecometoourknowledge1butalsotoextendthetextatvariousplaces.Thismainlyconcernstheroleofthefinitedifferenceandthefinitevolumemethods,whichhavegainedmoreattentionbyaslightextensionofChapters1and6andbyaconsiderableextensionofChapter7.Time-dependentproblemsarenowtreatedwithallthreeapproaches(fi-nitedifferences,finiteelements,andfinitevolumes),doingthisinauniformwayasfaraspossible.ThisalsomadeareorderingofChapters6–8nec-essary.Also,theindexhasbeenenlarged.Toimprovethedirectusabilityincourses,exercisesnowfolloweachsectionandshouldprovideenoughmaterialforhomework.Thisnewversionofthebookwouldnothavecomeintoexistencewithoutouralreadymentionedteamofhelpers,whoalsocarriedoutfirstversionsoftranslationsofpartsofthebook.Beyondthosealreadymentioned,theteamwasenforcedbyCeciliaDavid,BascaJadamba,Dr.SergeKr¨autle,Dr.WilhelmMerz,andPeterMirsch.AlexanderPrechtelnowtookchargeofthedifficultmodificationprocess.Prof.PaulDuChateausuggestedim-provements.Wewanttoextendourgratitudetoallofthem.Finally,we1UsersoftheGermaneditionmayconsulthttp://www.math.tu-clausthal.de/˜mala/publications/errata.pdf viiiPrefacetotheEnglishEditionthanksenioreditorAchiDosanjh,fromSpringer-VerlagNewYork,Inc.,forherconstantencouragement.RemarksfortheReaderandtheUseinLecturesThesizeofthetextcorrespondsroughlytofourhoursoflecturesperweekovertwoterms.Ifthecourselastsonlyoneterm,thenaselectionisnec-essary,whichshouldbeorientatedtotheaudience.Werecommendthefollowing“cuts”:Chapter0maybeskippedifthepartialdifferentialequationstreatedthereinarefamiliar.Section0.5shouldbeconsultedbecauseofthenotationcollectedthere.ThesameistrueforChapter1;possiblySection1.4maybeintegratedintoChapter3ifonewantstodealwithSection3.9orwithSection7.5.Chapters2and3arethecoreofthebook.Theinductivepresenta-tionthatwepreferredforsometheoreticalaspectsmaybeshortenedforstudentsofmathematics.Tothelecturer’stasteanddependingontheknowledgeoftheaudienceinnumericalmathematicsSection2.5maybeskipped.ThismightimpedethetreatmentoftheILUpreconditioninginSection5.3.ObservethatinSections2.1–2.3thetreatmentofthemodelproblemismergedwithbasicabstractstatements.Skippingthetreatmentofthemodelproblem,inturn,requiresanintegrationofthesestatementsintoChapter3.IndoingsoSection2.4maybeeasilycombinedwithSec-tion3.5.InChapter3thetheoreticalkernelconsistsofSections3.1,3.2.1,3.3–3.4.Chapter4presentsanoverviewofitssubject,notadetaileddevelopment,andisanextensionoftheclassicalsubjects,asareChapters6and9andtherelatedpartsofChapter7.IntheextensiveChapter5onemightfocusonspecialsubjectsorjustcon-siderSections5.2,5.3(and5.4)inordertopresentatleastonepracticallyrelevantandmoderniterativemethod.Section8.1andthefirstpartofSection8.2containbasicknowledgeofnumericalmathematicsand,dependingontheaudience,maybeomitted.Theappendicesaremeantonlyforconsultationandmaycompletethebasiclectures,suchasinanalysis,linearalgebra,andadvancedmathematicsforengineers.Concerningrelatedtextbooksforsupplementaryuse,tothebestofourknowledgethereisnonecoveringapproximatelythesametopics.Quiteafewdealwithfiniteelementmethods,andtheclosestoneinspiritprobablyis[21],butalso[6]or[7]haveacertainoverlap,andalsoofferadditionalmaterialnotcoveredhere.Fromthebooksspecialisedinfinitedifferencemethods,wemention[32]asanexample.The(node-oriented)finitevolumemethodispopularinengineering,inparticularinfluiddynamics,buttothebestofourknowledgethereisnopresentationsimilartooursina PrefacetotheEnglishEditionixmathematicaltextbook.ReferencestotextbooksspecialisedinthetopicsofChapters4,5and8aregiventhere.RemarksontheNotationPrintinginitalicsemphasizesdefinitionsofnotation,evenifthisisnotcarriedoutasanumbereddefinition.Vectorsappearindifferentforms:Besidesthe“short”spacevectorsx∈Rdthereare“long”representationvectorsu∈Rm,whichdescribeingeneralthedegreesoffreedomofafiniteelement(orvolume)approxi-mationorrepresentthevaluesongridpointsofafinitedifferencemethod.Herewechooseboldtype,alsoinordertohaveadistinctivefeaturefromthegeneratedfunctions,whichfrequentlyhavethesamenotation,orfromthegridfunctions.DeviationscanbefoundinChapter0,wherevectorialquantitiesbelong-ingtoRdareboldlytyped,andinChapters5and8,wheretheunknownsoflinearandnonlinearsystemsofequations,whicharetreatedinageneralmannerthere,aredenotedbyx∈Rm.Componentsofvectorswillbedesignatedbyasubindex,creatingadoubleindexforindexedquantities.Sequencesofvectorswillbesuppliedwithasuperindex(inparentheses);onlyinanabstractsettingdoweusesubindices.Erlangen,GermanyPeterKnabnerClausthal-Zellerfeld,GermanyLutzAngermannJanuary2002 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PrefacetotheGermanEditionThisbookresultedfromlecturesgivenattheUniversityofErlangen–NurembergandattheUniversityofMagdeburg.Ontheseoccasionsweoftenhadtodealwiththeproblemofaheterogeneousaudiencecomposedofstudentsofmathematicsandofdifferentnaturalorengineeringsciences.Thustheexpectationsofthestudentsconcerningthemathematicalaccu-racyandtheapplicabilityoftheresultswerewidelyspread.Ontheotherhand,neitherrelevantmodelsofpartialdifferentialequationsnorsomeknowledgeofthe(modern)theoryofpartialdifferentialequationscouldbeassumedamongthewholeaudience.Consequently,inordertoovercomethegivensituation,wehavechosenaselectionofmodelsandmethodsrelevantforapplications(whichmightbeextended)andattemptedtoilluminatethewholespectrum,extendingfromthetheorytotheimplementation,with-outassumingadvancedmathematicalbackground.Mostofthetheoreticalobstacles,difficultfornonmathematicians,willbetreatedinan“induc-tive”manner.Ingeneral,weuseanexplanatorystylewithout(hopefully)compromisingthemathematicalaccuracy.Wehopetosupplyespeciallystudentsofmathematicswiththein-formationnecessaryforthecomprehensionandimplementationoffiniteelement/finitevolumemethods.Forstudentsofthevariousnaturalorengineeringsciencesthetextoffers,beyondthepossiblyalreadyexistingknowledgeconcerningtheapplicationofthemethodsinspecialfields,anintroductionintothemathematicalfoundations,whichshouldfacilitatethetransformationofspecificknowledgetootherfieldsofapplications.Wewanttoexpressourgratitudeforthevaluablehelpthatwereceivedduringthewritingofthisbook:Dr.MarkusBause,SandroBitterlich, xiiPrefacetotheGermanEditionDr.ChristofEck,AlexanderPrechtel,JoachimRang,andDr.EckhardSchneiddidtheproofreadingandsuggestedimportantimprovements.Fromtheanonymousrefereeswereceivedusefulcomments.VeryspecialthanksgotoMrs.MagdalenaIhleandDr.GerhardSumm.Mrs.IhletransposedthetextquicklyandpreciselyintoTEX.Dr.SummnotonlyworkedontheoriginalscriptandontheTEX-form,healsoorganizedthecomplexanddistributedrewritingandextensionprocedure.Theeliminationofmanyinconsistenciesisduetohim.AdditionallyheinfluencedpartsofSec-tions3.4and3.8byhisoutstandingdiplomathesis.WealsowanttothankDr.ChistophTappforthepreparationofthegraphicofthetitleandforprovidingothergraphicsfromhisdoctoralthesis[70].Ofcourse,hintsconcerning(typing)mistakesandgeneralimprovementsarealwayswelcome.WethankSpringer-Verlagfortheirconstructivecollaboration.Last,butnotleast,wewanttoexpressourgratitudetoourfamiliesfortheirunderstandingandforbearance,whichwerenecessaryforusespeciallyduringthelastmonthsofwriting.Erlangen,GermanyPeterKnabnerMagdeburg,GermanyLutzAngermannFebruary2000 ContentsSeriesPrefacevPrefacetotheEnglishEditionviiPrefacetotheGermanEditionxi0ForExample:ModellingProcessesinPorousMediawithDifferentialEquations10.1TheBasicPartialDifferentialEquationModels.....10.2ReactionsandTransportinPorousMedia........50.3FluidFlowinPorousMedia................70.4ReactiveSoluteTransportinPorousMedia........110.5BoundaryandInitialValueProblems...........141FortheBeginning:TheFiniteDifferenceMethodforthePoissonEquation191.1TheDirichletProblemforthePoissonEquation.....191.2TheFiniteDifferenceMethod...............211.3GeneralizationsandLimitationsoftheFiniteDifferenceMethod..............291.4MaximumPrinciplesandStability.............362TheFiniteElementMethodforthePoissonEquation462.1VariationalFormulationfortheModelProblem.....46 xivContents2.2TheFiniteElementMethodwithLinearElements....552.3StabilityandConvergenceoftheFiniteElementMethod...................682.4TheImplementationoftheFiniteElementMethod:Part1............................742.5SolvingSparseSystemsofLinearEquationsbyDirectMethods.....................823TheFiniteElementMethodforLinearEllipticBoundaryValueProblemsofSecondOrder923.1VariationalEquationsandSobolevSpaces........923.2EllipticBoundaryValueProblemsofSecondOrder...1003.3ElementTypesandAffineEquivalentTriangulations.................1143.4ConvergenceRateEstimates................1313.5TheImplementationoftheFiniteElementMethod:Part2............................1483.6ConvergenceRateResultsinCaseofQuadratureandInterpolation...............1553.7TheConditionNumberofFiniteElementMatrices...1633.8GeneralDomainsandIsoparametricElements......1673.9TheMaximumPrincipleforFiniteElementMethods..1714GridGenerationandAPosterioriErrorEstimation1764.1GridGeneration.......................1764.2APosterioriErrorEstimatesandGridAdaptation...1855IterativeMethodsforSystemsofLinearEquations1985.1LinearStationaryIterativeMethods............2005.2GradientandConjugateGradientMethods........2175.3PreconditionedConjugateGradientMethod.......2275.4KrylovSubspaceMethodsforNonsymmetricSystemsofEquations.........2335.5TheMultigridMethod...................2385.6NestedIterations......................2516TheFiniteVolumeMethod2556.1TheBasicIdeaoftheFiniteVolumeMethod.......2566.2TheFiniteVolumeMethodforLinearEllipticDifferen-tialEquationsofSecondOrderonTriangularGrids...2627DiscretizationMethodsforParabolicInitialBoundaryValueProblems2837.1ProblemSettingandSolutionConcept..........2837.2SemidiscretizationbytheVerticalMethodofLines...293 Contentsxv7.3FullyDiscreteSchemes...................3117.4Stability...........................3157.5TheMaximumPrinciplefortheOne-Step-ThetaMethod..................3237.6OrderofConvergenceEstimates..............3308IterativeMethodsforNonlinearEquations3428.1Fixed-PointIterations....................3448.2Newton’sMethodandItsVariants............3488.3SemilinearBoundaryValueProblemsforEllipticandParabolicEquations..................3609DiscretizationMethodsforConvection-DominatedProblems3689.1StandardMethodsandConvection-DominatedProblems.............3689.2TheStreamline-DiffusionMethod.............3759.3FiniteVolumeMethods...................3839.4TheLagrange–GalerkinMethod..............387AAppendices390A.1Notation...........................390A.2BasicConceptsofAnalysis.................393A.3BasicConceptsofLinearAlgebra.............394A.4SomeDefinitionsandArgumentsofLinearFunctionalAnalysis.....................399A.5FunctionSpaces.......................404References:TextbooksandMonographs409References:JournalPapers412Index415 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0ForExample:ModellingProcessesinPorousMediawithDifferentialEquationsThischapterillustratesthescientificcontextinwhichdifferentialequationmodelsmayoccur,ingeneral,andalsoinaspecificexample.Section0.1reviewsthefundamentalequations,forsomeofthemdiscretizationtech-niqueswillbedevelopedandinvestigatedinthisbook.InSections0.2–0.4wefocusonreactionandtransportprocessesinporousmedia.Thesesectionsareindependentoftheremainingpartsandmaybeskippedbythereader.Section0.5,however,shouldbeconsultedbecauseitfixessomenotationtobeusedlateron.0.1TheBasicPartialDifferentialEquationModelsPartialdifferentialequationsareequationsinvolvingsomepartialderiva-tivesofanunknownfunctionuinseveralindependentvariables.Partialdifferentialequationswhicharisefromthemodellingofspatial(andtempo-ral)processesinnatureortechnologyareofparticularinterest.Therefore,weassumethatthevariablesofuarex=(x,...,x)T∈Rdford≥1,1drepresentingaspatialpoint,andpossiblyt∈R,representingtime.Thustheminimalsetofvariablesis(x1,x2)or(x1,t),otherwisewehaveordinarydifferentialequations.Wewillassumethatx∈Ω,whereΩisaboundeddomain,e.g.,ametalworkpiece,oragroundwateraquifer,andt∈(0,T]forsome(timehorizon)T>0.NeverthelessalsoprocessesactinginthewholeRd×R,orinunboundedsubsetsofit,areofinterest.OnemayconsulttheAppendixfornotationsfromanalysisetc.usedhere.Oftenthefunctionu 20.ModellingProcessesinPorousMediawithDifferentialEquationsrepresents,orisrelatedto,thevolumedensityofanextensivequantitylikemass,energy,ormomentum,whichisconserved.IntheiroriginalformallquantitieshavedimensionsthatwedenoteinaccordancewiththeInter-nationalSystemofUnits(SI)andwriteinsquarebrackets[].Letabeasymbolfortheunitoftheextensivequantity,thenitsvolumedensityisassumedtohavetheformS=S(u),i.e.,theunitofS(u)isa/m3.Forexample,formassconservationa=kg,andS(u)isaconcentration.Fordescribingtheconservationweconsideranarbitrary“nottoobad”sub-setΩ˜⊂Ω,thecontrolvolume.ThetimevariationofthetotalextensivequantityinΩisthen˜
∂tS(u(x,t))dx.(0.1)Ω˜Ifthisfunctiondoesnotvanish,onlytworeasonsarepossibleduetocon-servation:—ThereisaninternallydistributedsourcedensityQ=Q(x,t,u)[a/m3/s],beingpositiveifS(u)isproduced,andnegativeifitisdestroyed,i.e.,onetermtobalance(0.1)isΩ˜Q(x,t,u(x,t))dx.—Thereisanetfluxoftheextensivequantityovertheboundary∂Ωof˜Ω˜.LetJ=J(x,t)[a/m2/s]denotethefluxdensity,i.e.,Jiistheamount,thatpassesaunitsquareperpendiculartotheithaxisinonesecondinthedirectionoftheithaxis(ifpositive),andintheoppositedirectionotherwise.Thenanothertermtobalance(0.1)isgivenby
−J(x,t)·ν(x)dσ,∂Ωwhereνdenotestheouterunitnormalon∂Ω.Summarizingtheconserva-tionreads


∂tS(u(x,t))dx=−J(x,t)·ν(x)dσ+Q(x,t,u(x,t))dx.(0.2)Ω˜∂Ω˜Ω˜TheintegraltheoremofGauss(see(2.3))andanexchangeoftimederivativeandintegralleadsto
[∂tS(u(x,t))+∇·J(x,t)−Q(x,t,u(x,t))]dx=0,Ω˜and,asΩisarbitrary,alsoto˜∂tS(u(x,t))+∇·J(x,t)=Q(x,t,u(x,t))forx∈Ω,t∈(0,T].(0.3)Allmanipulationshereareformalassumingthatthefunctionsinvolvedhavethenecessaryproperties.Thepartialdifferentialequation(0.3)isthebasicpointwiseconservationequation,(0.2)itscorrespondingintegralform.Equation(0.3)isonerequirementforthetwounknownsuandJ,thusit 0.1.TheBasicPartialDifferentialEquationModels3hastobeclosedbya(phenomenological)constitutivelaw,postulatingarelationbetweenJandu.AssumeΩisacontainerfilledwithafluidinwhichasubstanceisdis-solved.Ifuistheconcentrationofthissubstance,thenS(u)=uanda=kg.ThedescriptionofJdependsontheprocessesinvolved.Ifthefluidisatrest,thenfluxisonlypossibleduetomoleculardiffusion,i.e.,afluxfromhightolowconcentrationsduetorandommotionofthedissolvedparticles.Experimentalevidenceleadsto(1)J=−K∇u(0.4)withaparameterK>0[m2/s],themoleculardiffusivity.Equation(0.4)iscalledFick’slaw.Inothersituations,likeheatconductioninasolid,asimilarmodeloccurs.Here,urepresentsthetemperature,andtheunderlyingprincipleisenergyconservation.TheconstitutivelawisFourier’slaw,whichalsohastheform(0.4),butasKisamaterialparameter,itmayvarywithspaceor,foranisotropicmaterials,beamatrixinsteadofascalar.Thusweobtainthediffusionequation∂tu−∇·(K∇u)=Q.(0.5)IfKisscalarandconstant—letK=1byscaling—,andf:=Qisindependentofu,theequationsimplifiesfurtherto∂tu−∆u=f,where∆u:=∇·(∇u).Wementionedalreadythatthisequationalsooccursinthemodellingofheatconduction,thereforethisequationor(0.5)isalsocalledtheheatequation.Ifthefluidisinmotionwitha(given)velocitycthen(forced)convectionoftheparticlestakesplace,beingdescribedby(2)J=uc,(0.6)i.e.,takingbothprocessesintoaccount,themodeltakestheformoftheconvection-diffusionequation∂tu−∇·(K∇u−cu)=Q.(0.7)TherelativestrengthofthetwoprocessesismeasuredbytheP´ecletnumber(definedinSection0.4).Ifconvectionisdominatingonemayignorediffusionandonlyconsiderthetransportequation∂tu+∇·(cu)=Q.(0.8)Thedifferentnatureofthetwoprocesseshastobereflectedinthemodels,therefore,adapteddiscretizationtechniqueswillbenecessary.Inthisbookwewillconsidermodelslike(0.7),usuallywithasignificantcontributionofdiffusion,andthecaseofdominatingconvectionisstudiedinChapter9.Thepureconvectivecaselike(0.8)willnotbetreated. 40.ModellingProcessesinPorousMediawithDifferentialEquationsInmoregeneralversionsof(0.7)∂tuisreplacedby∂tS(u),whereSdependslinearlyornonlinearlyonu.InthecaseofheatconductionSistheinternalenergydensity,whichisrelatedtothetemperatureuviathefactorsmassdensityandspecificheat.Forsomematerialsthespecificheatdependsonthetemperature,thenSisanonlinearfunctionofu.FurtheraspectscomeintoplaybythesourcetermQifitdependslinearlyornonlinearlyonu,inparticulardueto(chemical)reactions.Examplesforthesecaseswillbedevelopedinthefollowingsections.Sinceequation(0.3)anditsexamplesdescribeconservationingeneral,itstillhastobeadaptedtoaconcretesituationtoensureauniquesolutionu.ThisisdonebythespecificationofaninitialconditionS(u(x,0))=S0(x)forx∈Ω,andbyboundaryconditions.Intheexampleofthewaterfilledcontainernomassfluxwilloccuracrossitswalls,therefore,thefollowingboundaryconditionJ·ν(x,t)=0forx∈∂Ω,t∈(0,T)(0.9)isappropriate,which—dependingonthedefinitionofJ—prescribesthenormalderivativeofu,oralinearcombinationofitandu.InSection0.5additionalsituationsaredepicted.Ifaprocessisstationary,i.e.time-independent,thenequation(0.3)reducesto∇·J(x)=Q(x,u(x))forx∈Ω,whichinthecaseofdiffusionandconvectionisspecifiedto−∇·(K∇u−cu)=Q.ForconstantK—letK=1byscaling—,c=0,andf:=Q,beingindependentofu,thisequationreducesto−∆u=finΩ,thePoissonequation.Insteadoftheboundarycondition(0.9),onecanprescribethevaluesofthefunctionuattheboundary:u(x)=g(x)forx∈∂Ω.Formodels,whereuisaconcentrationortemperature,thephysicalreali-sationofsuchaboundaryconditionmayraisequestions,butinmechanicalmodels,whereuistointerpretedasadisplacement,suchaboundarycon-ditionseemsreasonable.Thelastboundaryvalueproblemwillbethefirstmodel,whosediscretizationwillbediscussedinChapters1and2.Finallyitshouldbenotedthatitisadvisabletonon-dimensionalisethefinalmodelbeforenumericalmethodsareapplied.Thismeansthatboththeindependentvariablesxi(andt),andthedependentoneu,arereplaced 0.2.ReactionsandTransportinPorousMedia5byxi/xi,ref,t/tref,andu/uref,wherexi,ref,tref,andurefarefixedreferencevaluesofthesamedimensionasxi,t,andu,respectively.Thesereferencevaluesareconsideredtobeoftypicalsizefortheproblemsunderinvestiga-tion.Thisprocedurehastwoadvantages:Ontheonehand,thetypicalsizeisnow1,suchthatthereisanabsolutescalefor(anerrorin)aquantitytobesmallorlarge.Ontheotherhand,ifthereferencevaluesarechosenappropriatelyareductioninthenumberofequationparameterslikeKandcin(0.7)mightbepossible,havingonlyfeweralgebraicexpressionsoftheoriginalmaterialparametersintheequation.Thisfacilitatesnumericalparameterstudies.0.2ReactionsandTransportinPorousMediaAporousmediumisaheterogeneousmaterialconsistingofasolidmatrixandaporespacecontainedtherein.Weconsidertheporespace(oftheporousmedium)asconnected;otherwise,thetransportoffluidsintheporespacewouldnotbepossible.Porousmediaoccurinnatureandman-ufacturedmaterials.Soilsandaquifersareexamplesingeosciences;porouscatalysts,chromatographiccolumns,andceramicfoamsplayimportantrolesinchemicalengineering.Eventhehumanskincanbeconsideredaporousmedium.Inthefollowingwefocusonapplicationsinthegeosciences.Thusweuseaterminologyreferringtothenaturalsoilasaporousmedium.Onthemicroorporescaleofasinglegrainorpore,i.e.,inarangeofµmtomm,thefluidsconstitutedifferentphasesinthethermodynamicsense.Thuswenamethissysteminthecaseofkfluidsincludingthesolidmatrixas(k+1)-phasesystemorwespeakofk-phaseflow.Wedistinguishthreeclassesoffluidswithdifferentaffinitiestothesolidmatrix.Theseareanaqueousphase,markedwiththeindex“w”forwater,anonaqueousphaseliquid(likeoilorgasolineasnaturalresourcesorcon-taminants),markedwiththeindex“o,”andagaseousphase,markedwiththeindex“g”(e.g.,soilair).Locally,atleastoneofthesephaseshasal-waystobepresent;duringatransientprocessphasescanlocallydisappearorbegenerated.Thesefluidphasesareinturnmixturesofseveralcom-ponents.Inapplicationsoftheearthsciences,forexample,wedonotdealwithpurewaterbutencounterdifferentspeciesintrueorcolloidalsolu-tioninthesolventwater.Thewiderangeofchemicalcomponentsincludesplantnutrients,mineralnutrientsfromsaltdomes,organicdecompositionproducts,andvariousorganicandinorganicchemicals.Thesesubstancesarenormallynotinert,butaresubjecttoreactionsandtransformationprocesses.Alongwithdiffusion,forcedconvectioninducedbythemotionofthefluidistheessentialdrivingmechanismforthetransportofsolutes.Butwealsoencounternaturalconvectionbythecouplingofthedynamicsofthesubstancetothefluidflow.Thedescriptionlevelatthemicroscale 60.ModellingProcessesinPorousMediawithDifferentialEquationsthatwehaveusedsofarisnotsuitableforprocessesatthelaboratoryortechnicalscale,whichtakeplaceinrangesofcmtom,orevenforprocessesinacatchmentareawithunitsofkm.Forthosemacroscalesnewmodelshavetobedeveloped,whichemergefromaveragingproceduresofthemod-elsonthemicroscale.Theremayalsoexistprincipaldifferencesamongthevariousmacroscalesthatletusexpectdifferentmodels,whicharisefromeachotherbyupscaling.Butthisaspectwillnotbeinvestigatedherefur-ther.Forthetransitionofmicrotomacroscalestheengineeringsciencesprovidetheheuristicmethodofvolumeaveraging,andmathematicstherigorous(butofonlylimiteduse)approachofhomogenization(see[36]or[19]).Noneofthetwopossibilitiescanbedepictedherecompletely.Wherenecessarywewillrefertovolumeaveragingfor(heuristic)motivation.LetΩ⊂Rdbethedomainofinterest.Allsubsequentconsiderationsareformalinthesensethattheadmissibilityoftheanalyticmanipulationsissupposed.Thiscanbeachievedbytheassumptionofsufficientsmoothnessforthecorrespondingfunctionsanddomains.LetV⊂Ωbeanadmissiblerepresentativeelementaryvolumeinthesenseofvolumeaveragingaroundapointx∈Ω.Typicallytheshapeandthesizeofarepresentativeelementaryvolumeareselectedinsuchamannerthattheaveragedvaluesofallgeometriccharacteristicsofthemicrostruc-tureoftheporespaceareindependentofthesizeofVbutdependonthelocationofthepointx.Thenweobtainforagivenvariableωαinthephaseα(aftercontinuationofωαwith0outsideofα)thecorrespondingmacroscopicquantities,assignedtothelocationx,astheextrinsicphaseaverage
1ωα:=ωα|V|Vorastheintrinsicphaseaverage
α1ωα:=ωα.|Vα|VαHereVαdenotesthesubsetofVcorrespondingtoα.Lett∈(0,T)bethetimeatwhichtheprocessisobserved.Thenotationx∈ΩmeansthevectorinCartesiancoordinates,whosecoordinatesarereferredtobyx,y,andz∈R.Despitethisambiguitythemeaningcanalwaysbeclearlyderivedfromthecontext.Lettheindex“s”(forsolid)standforthesolidphase;thenφ(x):=|VVs||V|>0denotestheporosity,andforeveryliquidphaseα,Sα(x,t):=|Vα||VVs|≥0 0.3.FluidFlowinPorousMedia7isthesaturationofthephaseα.Herewesupposethatthesolidphaseisstableandimmobile.Thusω=φSωααααforafluidphaseαandSα=1.(0.10)α:fluidSoifthefluidphasesareimmiscibleonthemicroscale,theymaybemiscibleonthemacroscale,andtheimmiscibilityonthemacroscaleisanadditionalassumptionforthemodel.Asinotherdisciplinesthedifferentialequationmodelsarederivedherefromconservationlawsfortheextensivequantitiesmass,impulse,anden-ergy,supplementedbyconstitutiverelationships,wherewewanttofocusonthemass.0.3FluidFlowinPorousMediaConsideraliquidphaseαonthemicroscale.Inthischapter,forclarity,wewrite“short”vectorsinRdalsoinboldwiththeexceptionofthecoordinatevectorx.Let˜α[kg/m3]bethe(microscopic)density,q˜:=˜ηv˜η˜ααη[m/s]themassaveragemixturevelocitybasedontheparticlevelocityv˜ηofacomponentηanditsconcentrationinsolution˜[kg/m3].ThetransportηtheoremofReynolds(see,forexample,[10])leadstothemassconservationlaw∂t˜α+∇·(˜αq˜α)=f˜α(0.11)withadistributedmasssourcedensityf˜α.Byaveragingweobtainfromherethemassconservationlaw∂t(φSαα)+∇·(αqα)=fα(0.12)withα,thedensityofphaseα,astheintrinsicphaseaverageof˜αandqα,thevolumetricfluidvelocityorDarcyvelocityofthephaseα,astheextrinsicphaseaverageofq˜α.Correspondingly,fαisanaveragemasssourcedensity.Beforeweproceedinthegeneraldiscussion,wewanttoconsidersomespecificsituations:Theareabetweenthegroundwatertableandtheimper-meablebodyofanaquiferischaracterizedbythefactthatthewholeporespaceisoccupiedbyafluidphase,thesoilwater.Thecorrespondingsatu-rationthusequals1everywhere,andwithomissionoftheindexequation(0.12)takestheform∂t(φ)+∇·(q)=f.(0.13) 80.ModellingProcessesinPorousMediawithDifferentialEquationsIfthedensityofwaterisassumedtobeconstant,duetoneglectingthemassofsolutesandcompressibilityofwater,equation(0.13)simplifiesfurthertothestationaryequation∇·q=f,(0.14)wherefhasbeenreplacedbythevolumesourcedensityf/,keepingthesamenotation.Thisequationwillbecompletedbyarelationshipthatcanbeinterpretedasthemacroscopicanalogueoftheconservationofmo-mentum,butshouldbeaccountedhereonlyasanexperimentallyderivedconstitutiverelationship.ThisrelationshipiscalledDarcy’slaw,whichreadsasq=−K(∇p+gez)(0.15)andcanbeappliedintherangeoflaminarflow.Herep[N/m2]istheintrinsicaverageofthewaterpressure,g[m/s2]thegravitationalacceleration,ethezunitvectorinthez-directionorientedagainstthegravitation,K=k/µ,(0.16)aquantity,whichisgivenbythepermeabilitykdeterminedbythesolidphase,andtheviscosityµdeterminedbythefluidphase.Forananisotropicsolid,thematrixk=k(x)isasymmetricpositivedefinitematrix.Inserting(0.15)in(0.14)andreplacingKbyKg,knownashydraulicconductivityintheliterature,andkeepingthesamenotationgivesthefollowinglinearequationfor1h(x,t):=p(x,t)+z,gthepiezometricheadh[m]:−∇·(K∇h)=f.(0.17)Theresultingequationisstationaryandlinear.Wecalladifferentialequa-tionmodelstationaryifitdependsonlyonthelocationxandnotonthetimet,andinstationaryotherwise.Adifferentialequationandcorrespond-ingboundaryconditions(cf.Section0.5)arecalledlinearifthesumorascalarmultipleofasolutionagainformsasolutionforthesum,respectivelythescalarmultiple,ofthesources.Ifwedealwithanisotropicsolidmatrix,wehaveK=KIwiththed×dunitmatrixIandascalarfunctionK.Equation(0.17)inthiscasereads−∇·(K∇h)=f.(0.18)Finallyifthesolidmatrixishomogeneous,i.e.,Kisconstant,wegetfromdivisionbyKandmaintainingthenotationfthePoissonequation−∆h=f,(0.19) 0.3.FluidFlowinPorousMedia9whichistermedtheLaplaceequationforf=0.Thismodelanditsmoregeneralformulationsoccurinvariouscontexts.If,contrarytotheaboveas-sumption,thesolidmatrixiscompressibleunderthepressureofthewater,andifwesuppose(0.13)tobevalid,thenwecanestablisharelationshipφ=φ(x,t)=φ0(x)φf(p)withφ(x)>0andamonotoneincreasingφsuchthatwithS(p):=φ
(p)0ffwegettheequationφ0S(p)∂tp+∇·q=fandtheinstationaryequationscorrespondingto(0.17)–(0.19),respectively.ForconstantS(p)>0thisyieldsthefollowinglinearequation:φ0S∂th−∇·(K∇h)=f,(0.20)whichalsorepresentsacommonmodelinmanycontextsandisknownfromcorrespondingfieldsofapplicationastheheatconductionequation.Weconsidersinglephaseflowfurther,butnowwewillconsidergasasfluidphase.Becauseofthecompressibility,thedensityisafunctionofthepressure,whichisinvertibleduetoitsstrictmonotonicitytop=P().Togetherwith(0.13)and(0.15)wegetanonlinearvariantoftheheatconductionequationintheunknown:∂(φ)−∇·K(∇P()+2ge)=f,(0.21)tzwhichalsocontainsderivativesoffirstorderinspace.IfP()=ln(α)holdsforaconstantα>0,then∇P()simplifiestoα∇.Thusforhorizontalflowweagainencountertheheatconductionequation.FortherelationshipP()=αsuggestedbytheuniversalgaslaw,α∇=1α∇2remains2nonlinear.Thechoiceofthevariableu:=2wouldresultinu1/2inthetimederivativeastheonlynonlinearity.Thusintheformulationinthecoefficientof∇disappearsinthedivergenceof=0.Correspondingly,thecoefficientS(u)=1φu−1/2of∂uintheformulationinubecomes2tunboundedforu=0.Inbothversionstheequationsaredegenerate,whosetreatmentisbeyondthescopeofthisbook.Avariantofthisequationhasgainedmuchattentionastheporousmediumequation(withconvection)inthefieldofanalysis(see,forexample,[42]).Returningtothegeneralframework,thefollowinggeneralizationofDarcy’slawcanbejustifiedexperimentallyforseveralliquidphases:krαqα=−k(∇pα+αgez).µαHeretherelativepermeabilitykrαofthephaseαdependsuponthesaturationsofthepresentphasesandtakesvaluesin[0,1]. 100.ModellingProcessesinPorousMediawithDifferentialEquationsAttheinterfaceoftwoliquidphasesα1andα2weobserveadifferenceofthepressures,theso-calledcapillarypressure,thatturnsoutexperimentallytobeafunctionofthesaturations:pcα1α2:=pα1−pα2=Fα1α2(Sw,So,Sg).(0.22)Ageneralmodelformultiphaseflow,formulatedforthemomentintermsofthevariablespα,Sα,isthusgivenbytheequations∂t(φSαα)−∇·(αλαk(∇pα+αgez))=fα(0.23)withthemobilitiesλα:=krα/µα,andtheequations(0.22)and(0.10),whereoneoftheSα’scanbeeliminated.Fortwoliquidphaseswandg,e.g.,waterandair,equations(0.22)and(0.10)forα=w,greadpc=pg−pw=F(Sw)andSg=1−Sw.Apparently,thisisatime-dependent,nonlinearmodelinthevariablespw,pg,Sw,whereoneofthevariablescanbeeliminated.Assumingconstantdensitiesα,furtherformulationsbasedon∇·qw+qg=fw/w+fg/g(0.24)canbegivenasconsequencesof(0.10).Theseequationsconsistofasta-tionaryequationforanewquantity,theglobalpressure,basedon(0.24),andatime-dependentequationforoneofthesaturations(seeExercise0.2).Inmanysituationsitisjustifiedtoassumeagaseousphasewithconstantpressureinthewholedomainandtoscalethispressuretopg=0.Thusforψ:=pw=−pcwehaveφ∂tS(ψ)−∇·(λ(ψ)k(∇ψ+gez))=fw/w(0.25)withconstantpressure:=,andS(ψ):=F−1(−ψ)asastrictlywmonotoneincreasingnonlinearityaswellasλ.Withtheconventiontosetthevalueoftheairpressureto0,thepressureintheaqueousphaseisintheunsaturatedstate,wherethegaseousphaseisalsopresent,andrepresentedbynegativevalues.Thewaterpressureψ=0marksthetransitionfromtheunsaturatedtothesaturatedzone.Thusintheunsaturatedzone,equation(0.25)representsanonlinearvariantoftheheatconductionequationforψ<0,theRichardsequation.AsmostfunctionalrelationshipshavethepropertyS
(0)=0,theequationdegeneratesintheabsenceofagaseousphase,namelytoastationaryequationinawaythatisdifferentfromabove.Equation(0.25)withS(ψ):=1andλ(ψ):=λ(0)canbecontinuedinaconsistentwaywith(0.14)and(0.15)alsoforψ≥0,i.e.,forthecaseofasoleaqueousphase.TheresultingequationisalsocalledRichardsequationoramodelofsaturated-unsaturatedflow. 0.4.ReactiveSoluteTransportinPorousMedia110.4ReactiveSoluteTransportinPorousMediaInthischapterwewilldiscussthetransportofasinglecomponentinaliquidphaseandsomeselectedreactions.Wewillalwaysrefertowaterasliquidphaseexplicitly.Althoughwetreatinhomogeneousreactionsintermsofsurfacereactionswiththesolidphase,wewanttoignoreexchangeprocessesbetweenthefluidphases.Onthemicroscopicscalethemasscon-servationlawforasinglecomponentηis,inthenotationof(0.11)byomittingthephaseindexw,∂t˜η+∇·(˜ηq˜)+∇·Jη=Q˜η,where2Jη:=˜η(v˜η−q˜)[kg/m/s](0.26)representsthediffusivemassfluxofthecomponentηandQ˜[kg/m3/s]isηitsvolumetricproductionrate.Foradescriptionofreactionsviathemassactionlawitisappropriatetochoosethemoleastheunitofmass.Thediffusivemassfluxrequiresaphenomenologicaldescription.Theassump-tionthatsolelybinarymoleculardiffusion,describedbyFick’slaw,actsbetweenthecomponentηandthesolvent,meansthatJη=−D˜η∇(˜η/˜)(0.27)withamoleculardiffusivityD>0[m2/s].Theaveragingprocedureappliedηon(0.26),(0.27)leadsto∂(Θc)+∇·(qc)+∇·J(1)+∇·J(2)=Q(1)+Q(2)tηηηηforthesoluteconcentrationofthecomponentη,c[kg/m3],asintrinsicη(1)(2)phaseaverageof˜η.Here,wehaveJastheaverageofJηandJ,themassfluxduetomechanicaldispersion,anewlyemergingtermatthemacroscopicscale.Analogously,Q(1)Q˜,ηistheintrinsicphaseaverageofη(2)andQηisanewlyemergingtermdescribingtheexchangebetweentheliquidandsolidphases.ThevolumetricwatercontentisgivenbyΘ:=φSwwiththewatersaturationSw.Experimentally,thefollowingphenomenologicaldescriptionsaresuggested:(1)J=−ΘτDη∇cηwithatortuosityfactorτ∈(0,1],(2)J=−ΘDmech∇cη,(0.28)andasymmetricpositivedefinitematrixofmechanicaldispersionDmech,whichdependsonq/Θ.Consequently,theresultingdifferentialequationreads∂t(Θcη)+∇·(qcη−ΘD∇cη)=Qη(0.29) 120.ModellingProcessesinPorousMediawithDifferentialEquations(1)(2)withD:=τDη+Dmech,Qη:=Qη+Qη.Becausethemassfluxconsistsofqcη,apartduetoforcedconvection,and(1)(2)ofJ+J,apartthatcorrespondstoageneralizedFick’slaw,anequa-tionlike(0.29)iscalledaconvection-diffusionequation.Accordingly,forthepartwithfirstspatialderivativeslike∇·(qcη)thetermconvectivepartisused,andforthepartwithsecondspatialderivativeslike−∇·(ΘD∇cη)thetermdiffusivepartisused.Ifthefirsttermdeterminesthecharacterofthesolution,theequationiscalledconvection-dominated.TheoccurrenceofsuchasituationismeasuredbythequantityPe,theglobalP´ecletnum-ber,thathastheformPe=qL/ΘD[-].HereLisacharacteristiclengthofthedomainΩ.Theextremecaseofpurelyconvectivetransportresultsinaconservationequationoffirstorder.Sincethecommonmod-elsforthedispersionmatrixleadtoaboundforPe,thereductiontothepurelyconvectivetransportisnotreasonable.However,wehavetotakeconvection-dominatedproblemsintoconsideration.Likewise,wespeakofdiffusivepartsin(0.17)and(0.20)andof(nonlin-ear)diffusiveandconvectivepartsin(0.21)and(0.25).Also,themultiphasetransportequationcanbeformulatedasanonlinearconvection-diffusionequationbyuseof(0.24)(seeExercise0.2),whereconvectionoftendom-inates.IftheproductionrateQηisindependentofcη,equation(0.29)islinear.Ingeneral,incaseofasurfacereactionofthecomponentη,thekineticsofthereactionhavetobedescribed.Ifthiscomponentisnotincompetitionwiththeothercomponents,onespeaksofadsorption.Thekineticequationthustakesthegeneralform∂tsη(x,t)=kηfη(x,cη(x,t),sη(x,t))(0.30)witharateparameterkηforthesorbedconcentrationsη[kg/kg],whichisgiveninreferencetothemassofthesolidmatrix.Here,thecomponentsinsorbedformareconsideredspatiallyimmobile.Theconservationofthetotalmassofthecomponentundergoingsorptiongives(2)Qη=−b∂tsη(0.31)withthebulkdensityb=s(1−φ),wheresdenotesthedensityofthesolidphase.With(0.30),(0.31)wehaveasystemconsistingofaninstationarypartialandanordinarydifferentialequation(withx∈Ωasparameter).AwidespreadmodelbyLangmuirreadsfη=kacη(sη−sη)−kdsηwithconstantska,kdthatdependuponthetemperature(amongotherfactors),andasaturationconcentrationsη(cf.forexample[24]).Ifweassumefη=fη(x,cη)forsimplicity,wegetascalarnonlinearequationincη,(1)∂t(Θcη)+∇·(qcη−ΘD∇cη)+bkηfη(·,cη)=Qη,(0.32) 0.4.ReactiveSoluteTransportinPorousMedia13andsηisdecoupledandextractedfrom(0.30).Ifthetimescalesoftransportandreactiondiffergreatly,andthelimitcasekη→∞isreasonable,then(0.30)isreplacedbyfη(x,cη(x,t),sη(x,t))=0.Ifthisequationissolvableforsη,i.e.,sη(x,t)=ϕη(x,cη(x,t)),thefollowingscalarequationforcηwithanonlinearityinthetimederivativeemerges:∂(Θc+ϕ(·,c))+∇·(qc−ΘD∇c)=Q(1).tηbηηηηηIfthecomponentηisincompetitionwithothercomponentsinthesur-facereaction,as,e.g.,inionexchange,thenfηhastobereplacedbyanonlinearitythatdependsontheconcentrationsofallinvolvedcomponentsc1,...,cN,s1,...,sN.Thusweobtainacoupledsysteminthesevariables.Finally,ifweencounterhomogeneousreactionsthattakeplacesolelyinthe(1)fluidphase,ananalogousstatementistrueforthesourcetermQη.Exercises0.1Giveageometricinterpretationforthematrixconditionofkin(0.16)andDmechin(0.28).0.2Considerthetwo-phaseflow(withconstantα,α∈{w,g})∂t(φSα)+∇·qα=fα,qα=−λαk(∇pα+αgez),Sw+Sg=1,pg−pw=pcwithcoefficientfunctionspc=pc(Sw),λα=λα(Sw),α∈{w,g}.Startingfromequation(0.23),performatransformationtothenewvariablesq=qw+qg,“totalflow,”
S11λg−λwdpcp=(pw+pg)+dξ,“globalpressure,”22Scλg+λwdξandthewatersaturationSw.Derivearepresentationofthephaseflowsinthenewvariables. 140.ModellingProcessesinPorousMediawithDifferentialEquations0.3AfrequentlyemployedmodelformechanicaldispersionisDmech=λL|v|2Pv+λT|v|2(I−Pv)withparametersλ>λ,wherev=q/ΘandPv=vvT/|v|2.HereLT2λLandλTarethelongitudinalandtransversaldispersionlengths.Giveageometricalinterpretation.0.5BoundaryandInitialValueProblemsThedifferentialequationsthatwederivedinSections0.3and0.4havethecommonform∂tS(u)+∇·(C(u)−K(∇u))=Q(u)(0.33)withasourcetermS,aconvectivepartC,adiffusivepartK,i.e.,atotalfluxC−KandasourcetermQ,whichdependlinearlyornonlinearlyontheunknownu.Forsimplification,weassumeutobeascalar.ThenonlinearitiesS,C,K,andQmayalsodependonxandt,whichshallbesuppressedinthenotationinthefollowing.Suchanequationissaidtobeindivergenceformorinconservativeform;amoregeneralformulationisobtainedbydifferentiating∇·C(u)=∂C(u)·∇u+(∇·C)(u)orby∂uintroducingageneralized“sourceterm”Q=Q(u,∇u).Uptonowwehaveconsidereddifferentialequationspointwiseinx∈Ω(andt∈(0,T))undertheassumptionthatalloccurringfunctionsarewell-defined.DuetotheapplicabilityoftheintegraltheoremofGaussonΩ˜⊂Ω(cf.(3.10)),theintegralformoftheconservationequationfollowsstraightforwardlyfromtheabove:


∂tS(u)dx+(C(u)−K(∇u))·νdσ=Q(u,∇u)dx(0.34)Ω˜∂Ω˜Ω˜withtheouterunitnormalν(seeTheorem3.8)forafixedtimetoralsointintegratedover(0,T).Indeed,thisequation(onthemicroscopicscale)istheprimarydescriptionoftheconservationofanextensivequantity:ChangesintimethroughstorageandsourcesinΩarecompensatedbythe˜normalfluxover∂Ω.Moreover,for˜∂tS,∇·(C−K),andQcontinuousontheclosureofΩ,(0.33)followsfrom(0.34).If,ontheotherhand,˜FisahyperplaneinΩwherethematerialpropertiesmayrapidlychange,the˜jumpcondition[(C(u)−K(∇u))·ν]=0(0.35)forafixedunitnormalνonFfollowsfrom(0.34),where[·]denotesthedifferenceoftheone-sidedlimits(seeExercise0.4).Sincethedifferentialequationdescribesconservationonlyingeneral,ithastobesupplementedbyinitialandboundaryconditionsinorderto 0.5.BoundaryandInitialValueProblems15specifyaparticularsituationwhereauniquesolutionisexpected.Boundaryconditionsarespecificationson∂Ω,whereνdenotestheouterunitnormal•ofthenormalcomponentoftheflux(inwards):−(C(u)−K(∇u))·ν=g1onΓ1(0.36)(fluxboundarycondition),•ofalinearcombinationofthenormalfluxandtheunknownitself:−(C(u)−K(∇u))·ν+αu=g2onΓ2(0.37)(mixedboundarycondition),•oftheunknownitself:u=g3onΓ3(0.38)(Dirichletboundarycondition).HereΓ1,Γ2,Γ3formadisjointdecompositionof∂Ω:∂Ω=Γ1∪Γ2∪Γ3,(0.39)whereΓ3issupposedtobeaclosedsubsetof∂Ω.Theinhomogeneitiesgiandthefactorαingeneraldependonx∈Ω,andfornonstationaryproblems(whereS(u)=0holds)ont∈(0,T).Theboundaryconditionsarelinearifthegidonotdepend(nonlinearly)onu(seebelow).Ifthegiarezero,wespeakofhomogeneous,otherwiseofinhomogeneous,boundaryconditions.Thusthepointwiseformulationofanonstationaryequation(whereSdoesnotvanish)requiresthevalidityoftheequationinthespace-timecylinderQT:=Ω×(0,T)andtheboundaryconditionsonthelateralsurfaceofthespace-timecylinderST:=∂Ω×(0,T).Differenttypesofboundaryconditionsarepossiblewithdecompositionsofthetype(0.39).Additionally,aninitialconditiononthebottomofthespace-timecylinderisnecessary:S(u(x,0))=S0(x)forx∈Ω.(0.40)Theseareso-calledinitial-boundaryvalueproblems;forstationaryprob-lemswespeakofboundaryvalueproblems.Asshownin(0.34)and(0.35)fluxboundaryconditionshaveanaturalrelationshipwiththedifferentialequation(0.33).ForalineardiffusivepartK(∇u)=K∇ualternativelywemayrequire∂νKu:=K∇u·ν=g1onΓ1,(0.41) 160.ModellingProcessesinPorousMediawithDifferentialEquationsandananalogousmixedboundarycondition.Thisboundaryconditionistheso-calledNeumannboundarycondition.SinceKissymmetric,∂νKu=∇u·Kνholds;i.e.,∂νKuisthederivativeindirectionoftheconormalKν.ForthespecialcaseK=Ithenormalderivativeisgiven.Incontrasttoordinarydifferentialequations,thereishardlyanygeneraltheoryofpartialdifferentialequations.Infact,wehavetodistinguishdif-ferenttypesofdifferentialequationsaccordingtothevariousdescribedphysicalphenomena.Thesedetermine,asdiscussed,different(initial-)boundaryvaluespecificationstorendertheproblemwell-posed.Well-posednessmeansthattheproblempossessesauniquesolution(withcertainpropertiesyettobedefined)thatdependscontinuously(inappropriatenorms)onthedataoftheproblem,inparticularonthe(initialand)boundaryvalues.Thereexistalsoill-posedboundaryvalueproblemsforpartialdifferentialequations,whichcorrespondtophysicalandtechnicalapplications.Theyrequirespecialtechniquesandshallnotbetreatedhere.Theclassificationintodifferenttypesissimpleiftheproblemislin-earandthedifferentialequationisofsecondorderasin(0.33).Byorderwemeanthehighestorderofthederivativewithrespecttothevariables(x1,...,xd,t)thatappears,wherethetimederivativeisconsideredtobelikeaspatialderivative.Almostalldifferentialequationstreatedinthisbookwillbeofsecondorder,althoughimportantmodelsinelasticitythe-oryareoffourthorderorcertaintransportphenomenaaremodelledbysystemsoffirstorder.Thedifferentialequation(0.33)isgenerallynonlinearduetothenonlin-earrelationshipsS,C,K,andQ.Suchanequationiscalledquasilinearifallderivativesofthehighestorderarelinear,i.e.,wehaveK(∇u)=K∇u(0.42)withamatrixK,whichmayalsodepend(nonlinearly)onx,t,andu.Furthermore,(0.33)iscalledsemilinearifnonlinearitiesarepresentonlyinu,butnotinthederivatives,i.e.,ifinadditionto(0.42)withKbeingindependentofu,wehaveS(u)=Su,C(u)=uc(0.43)withscalarandvectorialfunctionsSandc,respectively,whichmaydependonxandt.Suchvariablefactorsstandingbeforeuordifferentialtermsarecalledcoefficientsingeneral.Finally,thedifferentialequationislinearifwehave,inadditiontotheaboverequirements,Q(u)=−ru+fwithfunctionsrandfofxandt.Inthecasef=0thelineardifferentialequationistermedhomoge-neous,otherwiseinhomogeneous.Alineardifferentialequationobeysthesuperpositionprinciple:Supposeu1andu2aresolutionsof(0.33)withthe 0.5.BoundaryandInitialValueProblems17sourcetermsf1andf2andotherwiseidenticalcoefficientfunctions.Thenu1+γu2isasolutionofthesamedifferentialequationwiththesourcetermf1+γf2forarbitraryγ∈R.Thesameholdsforlinearboundaryconditions.Thetermsolutionofan(initial-)boundaryvalueproblemisusedhereinaclassicalsense,yettobespecified,whereallthequantitiesoccurringshouldsatisfypointwisecertainregularityconditions(seeDefini-tion1.1forthePoissonequation).However,forvariationalsolutions(seeDefinition2.2),whichareappropriateintheframeworkoffiniteelementmethods,theabovestatementsarealsovalid.Lineardifferentialequationsofsecondorderintwovariables(x,y)(in-cludingpossiblythetimevariable)canbeclassifiedindifferenttypesasfollows:Tothehomogeneousdifferentialequation∂2∂2∂2Lu=a(x,y)u+b(x,y)u+c(x,y)u∂x2∂x∂y∂y2(0.44)∂∂+d(x,y)u+e(x,y)u+f(x,y)u=0∂x∂ythefollowingquadraticformisassigned:(ξ,η)→a(x,y)ξ2+b(x,y)ξη+c(x,y)η2.(0.45)Accordingtoitseigenvalues,i.e.,theeigenvaluesofthematrixa(x,y)1b(x,y)2,(0.46)1b(x,y)c(x,y)2weclassifythetypes.Inanalogywiththeclassificationofconicsections,whicharedescribedby(0.45)(forfixed(x,y)),thedifferentialequation(0.44)iscalledatthepoint(x,y)•ellipticiftheeigenvaluesof(0.46)arenot0andhavethesamesign,•hyperbolicifoneeigenvalueispositiveandtheotherisnegative,•parabolicifexactlyoneeigenvalueisequalto0.Forthecorrespondinggeneralizationofthetermsford+1variablesandarbitraryorder,thestationaryboundaryvalueproblemswetreatinthisbookwillbeelliptic,ofsecondorder,and—exceptinChapter8—alsolinear;thenonstationaryinitial-boundaryvalueproblemswillbeparabolic.Systemsofhyperbolicdifferentialequationsoffirstorderrequirepartic-ularapproaches,whicharebeyondthescopeofthisbook.Nevertheless,wededicateChapter9toconvection-dominatedproblems,i.e.,ellipticorparabolicproblemsclosetothehyperboliclimitcase.Thedifferentdiscretizationstrategiesarebasedonvariousformulationsofthe(initial-)boundaryvalueproblems:Thefinitedifferencemethod,whichispresentedinSection1,andfurtheroutlinedfornonstationaryprob-lemsinChapter7,hasthepointwiseformulationof(0.33),(0.36)–(0.38) 180.ModellingProcessesinPorousMediawithDifferentialEquations(and(0.40))asastartingpoint.Thefiniteelementmethod,,whichliesinthefocusofourbook(Chapters2,3,and7),isbasedonanintegralformu-lationof(0.33)(whichwestillhavetodepict)thatincorporates(0.36)and(0.37).Theconditions(0.38)and(0.40)havetobeenforcedadditionally.Finally,thefinitevolumemethod(Chapters6and7)willbederivedfromtheintegralformulation(0.34),wherealsoinitialandboundaryconditionscomealongasinthefiniteelementapproach.Exercises0.4Derive(formally)(0.35)from(0.34).0.5Derivetheordersofthegivendifferentialoperatorsanddiffer-entialequations,anddecideineverycasewhethertheoperatorislinearornonlinear,andwhetherthelinearequationishomogeneousorinhomogeneous:(a)Lu:=uxx+xuy,(b)Lu:=ux+uuy,√(c)Lu:=1+x2(cosy)u+u−arctanxu=ln(x2+y2),xyxyy√(d)Lu:=ut+uxxxx+1+u=0,(e)u−u+x2=0.ttxx0.6(a)Determinethetypeofthegivendifferentialoperator:(i)Lu:=uxx−uxy+2uy+uyy−3uyx+4u,(ii)Lu=9uxx+6uxy+uyy+ux.(b)DeterminethepartsoftheplanewherethedifferentialoperatorLu:=yuxx−2uxy+xuyyiselliptic,hyperbolic,orparabolic.(c)(i)DeterminethetypeofLu:=3uy+uxy.(ii)ComputethegeneralsolutionofLu=0.0.7ConsidertheequationLu=fwithalineardifferentialoperatorofsecondorder,definedforfunctionsindvariables(d∈N)inx∈Ω⊂Rd.ThetransformationΦ:Ω→Ω
⊂Rdhasacontinuouslydifferentiable,nonsingularJacobimatrixDΦ:=∂Φ.∂xShowthatthepartialdifferentialequationdoesnotchangeitstypeifitiswritteninthenewcoordinatesξ=Φ(x). 1FortheBeginning:TheFiniteDifferenceMethodforthePoissonEquation1.1TheDirichletProblemforthePoissonEquationInthissectionwewanttointroducethefinitedifferencemethodusingthePoissonequationonarectangleasanexample.Bymeansofthisex-ampleandgeneralizationsoftheproblem,advantagesandlimitationsoftheapproachwillbeelucidated.Also,inthefollowingsectionthePoissonequationwillbethemaintopic,butthenonanarbitrarydomain.ForthespatialbasicsetofthedifferentialequationΩ⊂RdweassumeasminimalrequirementthatΩisadomain,whereadomainisanonempty,open,andconnectedset.Theboundaryofthisdomainwillbedenotedby∂Ω,theclosureΩ∪∂ΩbyΩ(seeAppendixA.2).TheDirichletproblemforthePoissonequationisthendefinedasfollows:Givenfunctionsg:∂Ω→Randf:Ω→R,wearelookingforafunctionu:Ω→Rsuchthatd∂2−u=finΩ,(1.1)∂x2i=1iu=gon∂Ω.(1.2)Thisdifferentialequationmodelhasalreadyappearedin(0.19)and(0.38)andbeyondthisapplicationhasanimportanceinawidespectrumofdisciplines.Theunknownfunctionucanbeinterpretedasanelectromag-neticpotential,adisplacementofanelasticmembrane,oratemperature.Similartothemulti-indexnotationtobeintroducedin(2.16)(butwith 201.FiniteDifferenceMethodforthePoissonEquationindicesatthetop)fromnowonforpartialderivativesweusethefollowingnotation.Notation:Foru:Ω⊂Rd→Rweset∂∂iu:=ufori=1,...,d,∂xi∂2∂iju:=ufori,j=1,···,d,∂xi∂xj∆u:=(∂11+...+∂dd)u.Theexpression∆uiscalledtheLaplaceoperator.Bymeansofthis,(1.1)canbewritteninabbreviatedformas−∆u=finΩ.(1.3)WecouldalsodefinetheLaplaceoperatorby∆u=∇·(∇u),Twhere∇u=(∂1u,...,∂du)denotesthegradientofafunctionu,and∇·v=∂1v1+···+∂dvdthedivergenceofavectorfieldv.Therefore,analternativenotationexists,whichwillnotbeusedinthefollowing:∆u=∇2u.Theincorporationoftheminussignintheleft-handsideof(1.3),whichlooksstrangeatfirstglance,isrelatedtothemonotonicityanddefinitenesspropertiesof−∆(seeSections1.4and2.1,respectively).Thenotionofasolutionfor(1.1),(1.2)stillhastospecifiedmorepre-cisely.Consideringtheequationsinapointwisesense,whichwillbepursuedinthischapter,thefunctionsin(1.1),(1.2)havetoexist,andtheequationshavetobesatisfiedpointwise.Since(1.1)isanequationonanopensetΩ,therearenoimplicationsforthebehaviourofuuptotheboundary∂Ω.Tohavearealrequirementduetotheboundarycondition,uhastobeatleastcontinuousuptotheboundary,thatis,onΩ.Theserequirementscanbeformulatedinacompactwaybymeansofcorrespondingfunctionspaces.ThefunctionspacesareintroducedmorepreciselyinAppendixA.5.SomeexamplesareC(Ω):=u:Ω→RucontinuousinΩ,C1(Ω):=u:Ω→Ru∈C(Ω),∂uexistsinΩ,i∂iu∈C(Ω)foralli=1,...,d.ThespacesCk(Ω)fork∈N,C(Ω),andCk(Ω),aswellasC(∂Ω),aredefinedanalogously.Ingeneral,therequirementsrelatedtothe(contin-uous)existenceofderivativesarecalled,alittlebitvaguely,smoothnessrequirements.Inthefollowing,inviewofthefinitedifferencemethod,fandgwillalsobeassumedcontinuousinΩand∂Ω,respectively. 1.2.TheFiniteDifferenceMethod21Definition1.1Assumef∈C(Ω)andg∈C(∂Ω).Afunctionuiscalleda(classical)solutionof(1.1),(1.2)ifu∈C2(Ω)∩C(Ω),(1.1)holdsforallx∈Ω,and(1.2)holdsforallx∈∂Ω.1.2TheFiniteDifferenceMethodThefinitedifferencemethodisbasedonthefollowingapproach:WearelookingforanapproximationtothesolutionofaboundaryvalueproblematafinitenumberofpointsinΩ(thegridpoints).Forthisreasonwesubstitutethederivativesin(1.1)bydifferencequotients,whichinvolveonlyfunctionvaluesatgridpointsinΩandrequire(1.2)onlyatgridpoints.Bythisweobtainalgebraicequationsfortheapproximatingvaluesatgridpoints.Ingeneral,suchaprocedureiscalledthediscretizationoftheboundaryvalueproblem.Sincetheboundaryvalueproblemislinear,thesystemofequationsfortheapproximatevaluesisalsolinear.Ingeneral,forother(differentialequation)problemsandotherdiscretizationapproacheswealsospeakofthediscreteproblemasanapproximationofthecontinuousproblem.Theaimoffurtherinvestigationswillbetoestimatetheresultingerrorandthustojudgethequalityoftheapproximativesolution.GenerationofGridPointsInthefollowing,forthebeginning,wewillrestrictourattentiontoproblemsintwospacedimensions(d=2).Forsimplificationweconsiderthecaseofaconstantstepsize(ormeshwidth)h>0inbothspacedirections.Thequantityhhereisthediscretizationparameter,whichinparticulardeterminesthedimensionofthediscreteproblem.◦◦◦◦◦◦◦◦◦•:Ωh◦•••••••◦l=8◦•••••••
◦◦:∂Ωh◦•••••••◦m=5
:farfromboundary◦•••••••◦◦◦◦◦◦◦◦◦◦:closetoboundaryFigure1.1.Gridpointsinasquaredomain.Forthetimebeing,letΩbearectangle,whichrepresentsthesimplestcaseforthefinitedifferencemethod(seeFigure1.1).BytranslationofthecoordinatesystemthesituationcanbereducedtoΩ=(0,a)×(0,b)witha,b>0.Weassumethatthelengthsa,b,andharesuchthata=lh,b=mhforcertainl,m∈N.(1.4) 221.FiniteDifferenceMethodforthePoissonEquationWedefineΩh:=(ih,jh)i=1,...,l−1,j=1,...,m−1(1.5)=(x,y)∈Ωx=ih,y=jhwithi,j∈ZasasetofgridpointsinΩinwhichanapproximationofthedifferentialequationhastobesatisfied.Inthesameway,∂Ωh:=(ih,jh)i∈{0,l},j∈{0,...,m}ori∈{0,...,l},j∈{0,m}=(x,y)∈∂Ωx=ih,y=jhwithi,j∈Zdefinesthegridpointson∂Ωinwhichanapproximationoftheboundaryconditionhastobesatisfied.TheunionofgridpointswillbedenotedbyΩh:=Ωh∪∂Ωh.SetupoftheSystemofEquationsLemma1.2LetΩ:=(x−h,x+h)forx∈R,h>0.ThenthereexistsaquantityR,dependingonuandh,theabsolutevalueofwhichcanbeboundedindependentlyofhandsuchthat(1)foru∈C2(Ω):
u(x+h)−u(x)1

u(x)=+hRand|R|≤u∞,h2(2)foru∈C2(Ω):
u(x)−u(x−h)1

u(x)=+hRand|R|≤u∞,h2(3)foru∈C3(Ω):
u(x+h)−u(x−h)21


u(x)=+hRand|R|≤u∞,2h6(4)foru∈C4(Ω):

u(x+h)−2u(x)+u(x−h)21(4)u(x)=+hRand|R|≤u∞.h212Herethemaximumnorm·∞(seeAppendixA.5)hastobetakenovertheintervaloftheinvolvedpoints(x,x+h),(x−h,x),or(x−h,x+h).Proof:TheprooffollowsimmediatelybyTaylorexpansion.Asanexampleweconsiderstatement3:Fromh2h3u(x±h)=u(x)±hu
(x)+u

(x)±u


(x±ξ)forcertainξ∈(0,h)±±26theassertionfollowsbylinearcombination.
1.2.DerivationandProperties23Notation:Thequotientinstatement1iscalledtheforwarddifferencequotient,anditisdenotedby∂+u(x).Thequotientinstatement2iscalledthebackwarddifferencequotient(∂−u(x)),andtheoneinstatement3thesymmetricdifferencequotient(∂0u(x)).Thequotientappearinginstatement4canbewrittenas∂−∂+u(x)bymeansoftheabovenotation.Inordertousestatement4ineveryspacedirectionfortheapproximationof∂11uand∂22uinagridpoint(ih,jh),inadditiontotheconditionsofDefinition1.1,thefurthersmoothnessproperties∂(3,0)u,∂(4,0)u∈C(Ω)andanalogouslyforthesecondcoordinatearenecessary.Hereweuse,e.g.,thenotation∂(3,0)u:=∂3u/∂x3(see(2.16)).1Usingtheseapproximationsfortheboundaryvalueproblem(1.1),(1.2),ateachgridpoint(ih,jh)∈Ωhwegetu((i+1)h,jh)−2u(ih,jh)+u((i−1)h,jh)−h2u(ih,(j+1)h)−2u(ih,jh)+u(ih,(j−1)h)+=(1.6)h2=f(ih,jh)+R(ih,jh)h2.HereRisasdescribedinstatement4ofLemma1.2,afunctiondependingonthesolutionuandonthestepsizeh,buttheabsolutevalueofwhichcanbeboundedindependentlyofh.Incaseswherewehavelesssmoothnessofthesolutionu,wecanneverthelessformulatetheapproximation(1.6)for−∆u,butthesizeoftheerrorintheequationisunclearatthemoment.Forthegridpoints(ih,jh)∈∂Ωhnoapproximationoftheboundaryconditionisnecessary:u(ih,jh)=g(ih,jh).IfweneglectthetermRh2in(1.6),wegetasystemoflinearequationsfortheapproximatingvaluesuijforu(x,y)atpoints(x,y)=(ih,jh)∈Ωh.Theyhavetheform12−ui,j−1−ui−1,j+4uij−ui+1,j−ui,j+1=fij(1.7)hfori=1,...,l−1,j=1,...,m−1,uij=gijifi∈{0,l},j=0,...,morj∈{0,m},i=0,...,l.(1.8)Hereweusedtheabbreviationsfij:=f(ih,jh),gij:=g(ih,jh).(1.9)Therefore,foreachunknowngridvalueuijwegetanequation.Thegridpoints(ih,jh)andtheapproximatingvaluesuijlocatedatthesehaveanaturaltwo-dimensionalindexing.Inequation(1.7)foragridpoint(i,j)onlytheneighboursatthefourcardinalpointsofthecompassappear,asitisdisplayedinFigure1.2.This 241.FiniteDifferenceMethodforthePoissonEquationinterconnectionisalsocalledthefive-pointstencilofthedifferencemethodandthemethodthefive-pointstencildiscretization.y(i,j+1)•(i−1,j)(i,j)(i+1,j)•••(i,j−1)•
xFigure1.2.Five-pointstencil.Attheinteriorgridpoints(x,y)=(ih,jh)∈Ωh,twocasescanbedistinguished:(1)(i,j)hasapositionsuchthatitsallneighbouringgridpointslieinΩh(farfromtheboundary).(2)(i,j)hasapositionsuchthatatleastoneneighbouringgridpoint(r,s)lieson∂Ωh(closetotheboundary).Theninequation(1.7)thevalueursisknowndueto(1.8)(urs=grs),and(1.7)canbemodifiedinthefollowingway:Removethevaluesurswith(rh,sh)∈∂Ωhintheequationsfor(i,j)closetotheboundaryandaddthevalueg/h2totheright-handrssideof(1.7).ThesetofequationsthatarisesbythiseliminationofboundaryunknownsbymeansofDirichletboundaryconditionswecall(1.7)∗;itisequivalentto(1.7),(1.8).Insteadofconsideringthevaluesuij,i=1,...,l−1,j=1,...,m−1,onealsospeaksofthegridfunctionuh:Ωh→R,whereuh(ih,jh)=uijfori=1,...,l−1,j=1,...,m−1.Gridfunctionson∂ΩhoronΩharedefinedanalogously.Thuswecanformulatethefinitedifferencemethodinthefollowingway:FindagridfunctionuhonΩhsuchthatequations(1.7),(1.8)hold,or,equivalentlyfindagridfunctionuhonΩhsuchthatequations(1.7)∗hold.StructureoftheSystemofEquationsAfterchoosinganorderingoftheuijfori=0,...,l,j=0,...,m,thesystemofequations(1.7)∗takesthefollowingform:Ahuh=qh(1.10)withA∈RM1,M1andu,q∈RM1,whereM=(l−1)(m−1).hhh1Thismeansthatnearlyidenticalnotationsforthegridfunctionanditsrepresentingvectorarechosenforafixednumberingofthegridpoints.Theonlydifferenceisthattherepresentingvectorisprintedinbold.Theorderingofthegridpointsmaybearbitrary,withtherestrictionthatthe 1.2.DerivationandProperties25pointsinΩhareenumeratedbythefirstM1indices,andthepointsin∂ΩharelabelledwiththesubsequentM2=2(l+m)indices.ThestructureofAhisnotinfluencedbythisrestriction.Becauseofthedescribedeliminationprocess,theright-handsideqhhasthefollowingform:qh=−Aˆhg+f,(1.11)whereg∈RM2andf∈RM1arethevectorsrepresentingthegridfunctionsfh:Ωh→Randgh:∂Ωh→Raccordingtothechosennumberingwiththevaluesdefinedin(1.9).ThematrixAˆ∈RM1,M2hasthefollowingform:h1−ifthenodeiisclosetotheboundaryh2(Aˆh)ij=andjisaneighbourinthefive-pointstencil,0otherwise.(1.12)Foranyordering,onlythediagonalelementandatmostfourfurtherentriesinarowofAh,definedby(1.7),aredifferentfrom0;thatis,thematrixissparseinastrictsense,asisassumedinChapter5.AnobviousorderingistherowwisenumberingofΩhaccordingtothefollowingscheme:(h,b−h)(2h,b−h)(a−h,b−h)······(l−1)(m−2)+1(l−1)(m−2)+2(l−1)(m−1)(h,b−2h)(2h,b−2h)(a−h,b−2h)······(l−1)(m−3)+1(l−1)(m−3)+2(l−1)(m−2).............(1.13)...(h,2h)(2h,2h)(a−h,2h)······ll+12l−2(h,h)(2h,h)(a−h,h)······12l−1Anothernameoftheaboveschemeislexicographicordering.(However,thisnameisbettersuitedtothecolumnwisenumbering.)InthiscasethematrixAhhasthefollowingformofan(m−1)×(m−1)blocktridiagonalmatrix:T−I−IT−I0.........A=h−2(1.14)h.........0−IT−I−IT 261.FiniteDifferenceMethodforthePoissonEquationwiththeunitmatrixI∈Rl−1,l−1and4−1−14−10.........T=∈Rl−1,l−1..........0−14−1−14Wereturntotheconsiderationofanarbitrarynumbering.Inthefol-lowingwecollectseveralpropertiesofthematrixA∈RM1,M1andthehextendedmatrixA˜:=AAˆ∈RM1,M,hhhwhereM:=M1+M2.ThematrixA˜htakesintoaccountallthegridpointsinΩh.Ithasnorelevancewiththeresolutionof(1.10),butwiththestabilityofthediscretization,whichwillbeinvestigatedinSection1.4.•(Ah)rr>0forallr=1,...,M1,•(A˜h)rs≤0forallr=1,...,M1,s=1,...,Msuchthatr=s,M1≥0forallr=1,...,M1,•(Ah)rsifrbelongstoagridpointcloseto(1.15)>0s=1theboundary,M•(A˜h)rs=0forallr=1,...,M1,s=1•Ahisirreducible,•Ahisregular.Therefore,thematrixAhisweaklyrowdiagonallydominant(seeAp-pendixA.3fordefinitionsfromlinearalgebra).Theirreducibilityfollowsfromthefactthattwoarbitrarygridpointsmaybeconnectedbyapathconsistingofcorrespondingneighboursinthefive-pointstencil.Thereg-ularityfollowsfromtheirreduciblediagonaldominance.Fromthiswecanconcludethat(1.10)canbesolvedbyGaussianeliminationwithoutpivotsearch.Inparticular,ifthematrixhasabandstructure,thiswillbepreserved.ThisfactwillbeexplainedinmoredetailinSection2.5.ThematrixAhhasthefollowingfurtherproperties:•Ahissymmetric,•Ahispositivedefinite.Itissufficienttoverifythesepropertiesforafixedordering,forexampletherowwiseone,sincebyachangeoftheorderingmatrix,AhistransformedtoPAPTwithsomeregularmatrixP,bywhichneithersymmetrynorh 1.2.DerivationandProperties27positivedefinitenessisdestroyed.Nevertheless,thesecondassertionisnotobvious.Onewaytoverifyitistocomputeeigenvaluesandeigenvectorsexplicitly,butwerefertoChapter2,wheretheassertionfollowsnaturallyfromLemma2.13and(2.36).Theeigenvaluesandeigenvectorsarespecifiedin(5.24)forthespecialcasel=m=nandalsoin(7.60).Therefore,(1.10)canberesolvedbyCholesky’smethod,takingintoaccountthebandedness.QualityoftheApproximationbytheFiniteDifferenceMethodWenowaddressthefollowingquestion:Towhataccuracydoesthegridfunctionuhcorrespondingtothesolutionuhof(1.10)approximatethesolutionuof(1.1),(1.2)?TothisendweconsiderthegridfunctionU:Ωh→R,whichisdefinedbyU(ih,jh):=u(ih,jh).(1.16)TomeasurethesizeofU−uh,weneedanorm(seeAppendixA.4andalsoA.5forthesubsequentlyuseddefinitions).Examplesarethemaximumnormuh−U∞:=max|(uh−U)(ih,jh)|(1.17)i=1,...,l−1j=1,...,m−1andthediscreteL2-norml−1m−11/22uh−U0,h:=h((uh−U)(ih,jh)).(1.18)i=1j=1Bothnormscanbeconceivedastheapplicationofthecontinuousnorms·ofthefunctionspaceL∞(Ω)or·ofthefunctionspaceL2(Ω)∞0topiecewiseconstantprolongationsofthegridfunctions(withaspecialtreatmentoftheareaclosetotheboundary).Obviously,wehave√vh0,h≤abvh∞foragridfunctionvh,butthereverseestimatedoesnotholduniformlyinh,sothat·∞isastrongernorm.Ingeneral,wearelookingforanorm·hinthespaceofgridfunctionsinwhichthemethodconvergesinthesenseuh−Uh→0forh→0orevenhasanorderofconvergencep>0,bywhichwemeantheexistenceofaconstantC>0independentofhsuchthatu−U≤Chp.hhDuetotheconstructionofthemethod,forasolutionu∈C4(Ω)wehave2AhU=qh+hR, 281.FiniteDifferenceMethodforthePoissonEquationwhereUandR∈RM1aretherepresentationsofthegridfunctionsUandRaccordingto(1.6)intheselectedordering.Therefore,wehave:A(u−U)=−h2Rhhandthus|A(u−U)|=h2|R|=Ch2hh∞∞withaconstantC(=|R|∞)>0independentofh.FromLemma1.2,4.weconcludethat1(4,0)(0,4)C=∂u∞+∂u∞.12Thatis,forasolutionu∈C4(Ω)themethodisconsistentwiththebound-aryvalueproblemwithanorderofconsistency2.Moregenerally,thenotiontakesthefollowingform:Definition1.3Let(1.10)bethesystemofequationsthatcorrespondstoa(finitedifference)approximationonthegridpointsΩhwithadiscretiza-tionparameterh.LetUbetherepresentationofthegridfunctionthatcorrespondstothesolutionuoftheboundaryvalueproblemaccordingto(1.16).Furthermore,let·hbeanorminthespaceofgridfunctionsonΩ,andlet|·|bethecorrespondingvectornorminthespaceRM1h,hhwhereM1histhenumberofgridpointsinΩh.Theapproximationiscalledconsistentwithrespectto·hif|AhU−qh|h→0forh→0.Theapproximationhastheorderofconsistencyp>0ifp|AhU−qh|h≤ChwithaconstantC>0independentofh.ThustheconsistencyortruncationerrorAhU−qhmeasuresthequalityofhowtheexactsolutionsatisfiestheapproximatingequations.Aswehaveseen,ingeneralitcanbedeterminedeasilybyTaylorexpansion,butattheexpenseofunnaturallyhighsmoothnessassumptions.Butonehastobecarefulinexpectingtheerror|uh−U|htobehaveliketheconsistencyerror.Wehaveu−U=A−1A(u−U)≤A−1A(u−U),(1.19)hhhhhhhhhhhwherethematrixnorm·hhastobechosentobecompatiblewiththevectornorm|·|h.TheerrorbehavesliketheconsistencyerrorasymptoticallyinhifA−1canbeboundedindependentlyofh;thatisifthemethodhhisstableinthefollowingsense:Definition1.4InthesituationofDefinition1.3,theapproximationiscalledstablewithrespectto·hifthereexistsaconstantC>0 Exercises29independentofhsuchthatA−1≤C.hhFromtheabovedefinitionwecanobviouslyconclude,with(1.19),thefollowingresult:Theorem1.5Aconsistentandstablemethodisconvergent,andtheorderofconvergenceisatleastequaltotheorderofconsistency.Therefore,specificallyforthefive-pointstencildiscretizationof(1.1),(1.2)onarectangle,stabilitywithrespectto·∞isdesirable.Infact,itfollowsfromthestructureofAh:Namely,wehave1A−1≤(a2+b2).(1.20)h∞16ThisfollowsfrommoregeneralconsiderationsinSection1.4(Theo-rem1.14).Puttingtheresultstogetherwehavethefollowingtheorem:Theorem1.6Letthesolutionuof(1.1),(1.2)onarectangleΩbeinC4(Ω).Thenthefive-pointstencildiscretizationhasanorderofconvergence2withrespectto·∞,moreprecisely,122(4,0)(0,4)2|uh−U|∞≤(a+b)∂u∞+∂u∞h.192Exercises1.1CompletetheproofofLemma1.2andalsoinvestigatetheerroroftherespectivedifferencequotients,assumingonlyu∈C2[x−h,x+h].1.2Generalizethediscussionconcerningthefive-pointstencildiscretiza-tion(includingtheorderofconvergence)of(1.1),(1.2)onarectangleforh1>0inthex1directionandh2>0inthex2direction.1.3Showthatanirreducibleweaklyrowdiagonallydominantmatrixcannothavevanishingdiagonalelements.1.3GeneralizationsandLimitationsoftheFiniteDifferenceMethodWecontinuetoconsidertheboundaryvalueproblem(1.1),(1.2)onarect-angleΩ.Thefive-pointstencildiscretizationdevelopedmaybeinterpretedasamapping−∆hfromfunctionsonΩhintogridfunctionsonΩh,which 301.FiniteDifferenceMethodforthePoissonEquationisdefinedby1−∆hvh(x1,x2):=cijvh(x1+ih,x2+jh),(1.21)i,j=−1wherec=4/h2,c=c=c=c=−1/h2,andc=0for0,00,11,00,−1−1,0ijallother(i,j).Forthedescriptionofsuchadifferencestencilasdefinedin(1.21)thepointsofthecompass(intwospacedimensions)mayalsobeinvolved.Inthefive-pointstencilonlythemainpointsofthecompassappear.Thequestionofwhethertheweightscijcanbechosendifferentlysuchthatwegainanapproximationof−∆uwithhigherorderinhhastobeanswerednegatively(seeExercise1.7).Inthisrespectthefive-pointstencilisoptimal.Thisdoesnotexcludethatotherdifferencestencilswithmoreentries,butofthesameorderofconvergence,mightbeworthwhiletocon-sider.Anexample,whichwillbederivedinExercise3.11bymeansofthefiniteelementmethod,hasthefollowingform:81c0,0=3h2,cij=−3h2forallotheri,j∈{−1,0,1}.(1.22)Thisnine-pointstencilcanbeinterpretedasalinearcombinationofthefive-pointstencilandafive-pointstencilforacoordinatesystemrotatedby√π(withstepsize2h),usingtheweights1and2inthislinearcombina-433tion.Usingageneralnine-pointstencilamethodwithorderofconsistencygreaterthan2canbeconstructedonlyiftheright-handsidefatthepoint(x1,x2)isapproximatednotbytheevaluationf(x1,x2),butbyapplyingamoregeneralstencil.Themehrstellenmethod(“Mehrstellenverfahren”)definedbyCollatzissuchanexample(see,forexample,[15,p.66]).Methodsofhigherordercanbeachievedbylargerstencils,meaningthatthesummationindicesin(1.21)havetobereplacedbykand−k,respectively,fork∈N.Butalreadyfork=2suchdifferencestencilscannotbeusedforgridpointsclosetotheboundary,sothatthereonehastoreturntoapproximationsoflowerorder.Ifweconsiderthefive-pointstenciltobeasuitablediscretizationforthePoissonequation,thehighsmoothnessassumptionforthesolutioninTheorem1.6shouldbenoted.Thisrequirementcannotbeignored,sinceingeneralitdoesnotholdtrue.Ontheonehand,forasmoothlyboundeddomain(seeAppendixA.5foradefinitionofadomainwithCl-boundary)thesmoothnessofthesolutionisdeterminedonlybythesmoothnessofthedatafandg(seeforexample[13,Theorem6.19]),butontheotherhand,cornersinthedomainreducethissmoothnessthemore,themorereentrantthecornersare.Letusconsiderthefollowingexamples:Fortheboundaryvalueproblem(1.1),(1.2)onarectangle(0,a)×(0,b)wechoosef=1andg=0;thismeansarbitrarilysmoothfunctions.Nevertheless,forthesolutionu,thestatementu∈C2(Ω)cannothold,becauseotherwise,−∆u(0,0)=1wouldbetrue,butontheotherhand, 1.3.GeneralizationsandLimitations31wehave∂1,1u(x,0)=0becauseoftheboundaryconditionandhencealso∂1,1u(0,0)=0and∂2,2u(0,y)=0analogously.Therefore,∂2,2u(0,0)=0.Consequently,−∆u(0,0)=0,whichcontradictstheassumptionabove.Therefore,Theorem1.6isnotapplicablehere.Inthesecondexampleweconsiderthedomainwithreentrantcorner(seeFigure1.3)Ω=(x,y)∈R2x2+y2<1,x<0ory>0.Ingeneral,ifweidentifyR2andC,thismeans(x,y)∈R2andz=x+iy∈C,wehavethatifw:C→Cisanalytic(holomorphic),thenboththerealandtheimaginarypartsw,w:C→Rareharmonic,whichmeansthattheysolve−∆u=0.yΩxFigure1.3.DomainΩwithreentrantcorner.Wechoosew(z):=z2/3.Thenthefunctionu(x,y):=(x+iy)2/3solvestheequation−∆u=0inΩ.Inpolarcoordinates,x=rcosϕ,y=rsinϕ,thefunctionutakestheformiϕ2/32/32u(x,y)=re=rsinϕ.3Therefore,usatisfiestheboundaryconditions23πueiϕ=sinϕfor0≤ϕ≤,(1.23)32u(x,y)=0otherwiseon∂Ω.Butnotethatw
(z)=2z−1/3isunboundedforz→0,sothat∂u,∂u312areunboundedfor(x,y)→0.Therefore,inthiscasewedonotevenhaveu∈C1(Ω).Theexamplesdonotshowthatthefive-pointstencildiscretizationisnotsuitablefortheboundaryvalueproblemsconsidered,buttheyshowthene-cessityofatheoryofconvergence,whichrequiresonlyasmuchsmoothnessaswastobeexpected.Inthefollowingwediscusssomegeneralizationsoftheboundaryvalueproblemsconsideredsofar. 321.FiniteDifferenceMethodforthePoissonEquationGeneralDomainsΩWecontinuetoconsider(1.1),(1.2)butonageneraldomaininR2,forwhichthepartsoftheboundaryarenotnecessarilyalignedtothecoor-dinateaxes.Thereforewecankeepthesecondequationin(1.5)asthedefinitionofΩh,buthavetoredefinethesetofboundarygridpoints∂Ωh.Forexample,ifforsomepoint(x,y)∈Ωhwehave(x−h,y)∈/Ω,thenthereexistsanumbers∈(0,1]suchthat(x−ϑh,y)∈Ωforallϑ∈[0,s)and(x−sh,y)∈/Ω.Then(x−sh,y)∈∂Ω,andthereforewedefine(x−sh,y)∈∂Ωh.Theothermainpointsofthecompassaretreatedanalogously.Inthiswaythegridspacinginthevicinityoftheboundarybecomesvariable;inparticular,itcanbesmallerthanh.Forthequalityoftheapproximationwehavethefollowingresult:Lemma1.7LetΩ=(x−h1,x+h2)forx∈R,h1,h2>0.(1)Thenforu∈C3(Ω),

2u(x+h2)−u(x)u(x)−u(x−h1)u(x)=−h1+h2h2h1+max{h1,h2}R,whereRisboundedindependentlyofh1,h2.(2)Therearenoα,β,γ∈Rsuchthatu

(x)=αu(x−h)+βu(x)+γu(x+h)+Rh2+Rh2121122forallpolynomialsuofdegree3ifh1=h2.Proof:Exercises1.4and1.5.
Thisleadstoadiscretizationthatisdifficulttosetupandforwhichtheorderofconsistencyandorderofconvergencearenoteasilydetermined.OtherBoundaryConditionsWewanttoconsiderthefollowingexample.Let∂Ω=Γ1∪Γ3bedividedintotwodisjointsubsets.Wearelookingforafunctionusuchthat−∆u=finΩ,∂νu:=∇u·ν=gonΓ1,(1.24)u=0onΓ3,whereν:∂Ω→Rdistheouterunitnormal,andthus∂uisthenormalνderivativeofu. 1.3.GeneralizationsandLimitations33Forapartoftheboundaryorientedinacoordinatedirection,∂νuisjustapositiveornegativepartialderivative.ButifonlygridpointsinΩaretobeused,only±∂+uand±∂−urespectively(inthecoordinateshorthogonaltothedirectionoftheboundary)areavailabledirectlyfromtheaboveapproximationswithacorrespondingreductionoftheorderofconsistency.Foraboundarypointwithouttheserestrictionsthequestionofhowtoapproximate∂νuappropriatelyisopen.Asanexampleweconsider(1.24)forarectangleΩ=(0,a)×(0,b),whereΓ1:={(a,y)|y∈(0,b)},Γ3:=ΓΓ1.(1.25)Attheboundarygridpoints(a,jh),j=1,...,m−1,∂2u=∇u·νisprescribed,whichcanbeapproximateddirectlyonlyby∂−u.DuetoLemma1.2,2thisleadstoareductionintheconsistencyorder(seeEx-ercise1.8).TheresultingsystemofequationsmayincludetheNeumannboundarygridpointsinthesetofunknowns,forwhichanequationwiththeentries1/hinthediagonaland−1/hinanoff-diagonalcorrespondingtotheeasternneighbour(a−h,jh)hastobeadded.Alternatively,thoseboundarypointscanbeeliminated,leadingfortheeasternneighbourtoamodifieddifferencestencil(multipliedbyh2)−1−13(1.26)−1fortheright-handsideh2f(a−h,jh)+hg(a,jh).Inbothcasesthematrixpropertiesofthesystemofequationsascollectedin(1.15)stillhold,withM1theexceptionofs=1(Ah)rs=0,bothfortheNeumannboundarypointsandtheirneighbours,ifnoDirichletboundarypointisinvolvedintheirstencil.Thustheterm“closetotheboundary”hastobeinterpretedas“closetotheDirichletboundary.”Ifonewantstotakeadvantageofthesymmetricdifferencequotient∂0u,then“artificial”valuesatnewexternalgridpoints(a+h,jh)appear.Tokeepthebalanceofunknownsandequations,itcanbeassumedthatthedifferentialequationalsoholdsat(a,jh),andthusitisdiscretizedwiththefive-pointstencilthere.Ifoneattributesthediscreteboundaryconditiontotheexternalgridpoint,thenagaintheproperties(1.15)holdwiththeabovementionedinterpretation.Alternatively,theexternalgridpointscanbeeliminated,leadingtoamodifieddifferencestencil(multipliedbyh2)at(a,jh):−1−24(1.27)−1fortheright-handsideh2f(a,jh)+2hg(a,jh),withthesameinterpretationofproperties(1.15). 341.FiniteDifferenceMethodforthePoissonEquationMoreGeneralDifferentialEquationsAsanexampleweconsiderthedifferentialequation−∇·(k∇u)=fonΩ(1.28)withacontinuouscoefficientfunctionk:Ω→R,whichisboundedfrombelowbyapositiveconstantonΩ.Thisequationstatestheconservationofanextensivequantityuwhosefluxis−k∇u(seeSection0.5).Thisshouldberespectedbythediscretization,andthereforetheformof(1.28)obtainedbyworkingoutthederivativesisnotrecommendedasabasisforthediscretization.Thedifferentialexpressionin(1.28)canbediscretizedbyasuccessiveapplicationofcentraldifferencequotients,butthenagaintheorderofconsistencyhastobeinvestigated.Inaddition,onehastotakeintoaccountthefactthatthesmoothnessofudependsonthesmoothnessofk.Ifprocessesinheterogeneousmaterialshavetobedescribed,thenkisoftendiscontinuous.Inthesimplestexamplekisassumedtotaketwodifferentvalues:LetΩ=Ω1∪Ω2andk|Ω1=k1>0,k|Ω2=k2>0withconstantsk1=k2.AsworkedoutinSection0.5,ontheinteriorboundaryS:=Ω1∩Ω2atransmissionconditionhastobeimposed:•uiscontinuous,•(k∇u)·νiscontinuous,whereνistheouternormalon∂Ω1,forexample.Thisleadstothefollowingconditionsonui,beingtherestrictionsofuonΩifori=1,2:−k1∆u1=finΩ1,(1.29)−k2∆u2=finΩ2,u1=u2onS,(1.30)k1∂νu1=k2∂νu2onS.Inthiscasethequestionofanappropriatediscretizationisalsoopen.Summarizing,wehavethefollowingcatalogueofrequirements:Wearelookingforanotionofsolutionfor(general)boundaryvalueproblemswithnonsmoothcoefficientsandright-handsidessuchthat,forexample,thetransmissionconditionisfulfilledautomatically.Wearelookingforadiscretizationongeneraldomainssuchthat,forexample,the(orderof)convergencecanalsobeassuredforlesssmoothsolutionsandalsoNeumannboundaryconditionsasin(1.24)canbetreatedeasily.Thefiniteelementmethodinthesubsequentchapterswillfulfiltheserequirementstoalargeextent. Exercises35Exercises1.4ProveLemma1.7,1.1.5Undertheassumptionthatu:Ω⊂R→Risasufficientlysmoothfunction,determineintheansatzαu(x−h1)+βu(x)+γu(x+h2),h1,h2>0,thecoefficientsα=α(h1,h2),β=β(h1,h2),γ=γ(h1,h2),suchthat(a)forx∈Ω,u
(x)willbeapproximatedwiththeorderashighaspossible,(b)forx∈Ω,u

(x)willbeapproximatedwiththeorderashighaspossible,andinparticular,prove1.7,2.Hint:Determinethecoefficientssuchthattheformulaisexactforpolynomialswiththedegreeashighaspossible.1.6LetΩ⊂R2beaboundeddomain.Forasufficientlysmoothfunctionu:Ω→Rdeterminethedifferenceformulawithanorderashighaspossibletoapproximate∂11u(x1,x2),usingthe9valuesu(x1+γ1h,x2+γ2h),whereγ1,γ2∈{−1,0,1}.1.7LetΩ⊂R2beaboundeddomain.Showthatin(1.21)thereexistsnochoiceofcijsuchthatforanarbitrarysmoothfunctionu:Ω→R,|∆u(x)−∆u(x)|≤Ch3hisvalidwithaconstantCindependentofh.1.8Fortheexample(1.24),(1.25),investigatetheorderofconsistencybothforthediscretization(1.26)and(1.27)inthemaximumnorm.ArethereimprovementspossibleconsideringthediscreteL2-norm?(See(1.18).)1.9Considerexample(1.24)withΓ1:={(a,y)|y∈(0,b)}∪{(x,b)|x∈(0,a]},Γ3:=ΓΓ1,anddiscusstheapplicabilityoftheone-sidedandthesymmetricdiffer-encequotientsfortheapproximationoftheNeumannboundarycondition,inparticularwithrespecttoproperties(1.15).Inwhichwaydoestheboundaryconditionat(a,b),wherenouniquenormalexists,havetobeinterpreted? 361.FiniteDifferenceMethodforthePoissonEquation1.10Generalizethediscussionconcerningthefive-pointstencildis-cretization(includingtheorderofconvergence)totheboundaryvalueproblem−∆u+ru=finΩ,u=gon∂Ω,forr>0andΩ:=(0,a)×(0,b).Toapproximatethereactivetermru,thefollowingschemesinthenotationof(1.21)aretobeused:(a)c0,0=1,cij=0otherwise,(b)c0,0>0,c0,1,c1,0,c0,−1,c−1,0≥0,cij=0otherwise,and1i,j=−1cij=1.1.4MaximumPrinciplesandStabilityInthissectiontheproofofthestabilityestimate(1.20),whichisstillmiss-ing,willbegiven.Forthisreasonwedevelopamoregeneralframework,inwhichwewillthenalsodiscussthefiniteelementmethod(seeSection3.9)andthetime-dependentproblems(seeSection7.5).Theboundaryvalueproblem(1.1),(1.2)satisfiesa(weak)maximumprincipleinthefollowingsense:Iffiscontinuousandf(x)≤0forallx∈Ω(forshortf≤0),thenmaxu(x)≤maxu(x).x∈Ωx∈∂ΩThismaximumprincipleisalsostronginthefollowingsense:Themaxi-mumofuonΩcanbeattainedinΩonlyifuisconstant(see,forexample,[13],alsoforthefollowingassertions).Byexchangingu,f,gby−u,−f,−g,respectively,weseethatthereisananalogous(strong)minimumprinciple.Thesameholdsformoregenerallineardifferentialequationsasin(1.28),whichmayalsocontainconvectiveparts(thismeansfirst-orderderiva-tives).Butiftheequationcontainsareactivepart(thismeanswithoutderivatives),asintheexample−∆u+ru=finΩwithacontinuousfunctionr:Ω→Rsuchthatr(x)≥0forx∈Ω,thereisaweakmaximumprincipleonlyinthefollowingform:Iff≤0,thenmaxu(x)≤maxmaxu(x),0.x∈Ωx∈∂ΩTheweakmaximumprincipledirectlyimpliesassertionsaboutthede-pendenceofthesolutionuoftheboundaryvalueproblemonthedatafandg;thismeansstabilityproperties.Onecanalsofollowthismethodininvestigatingthediscretization.Forthebasicexamplewehave 1.4.MaximumPrinciplesandStability37Theorem1.8LetuhbeagridfunctiononΩhdefinedby(1.7),(1.8)andsupposefij≤0foralli=1,...,l−1,j=1,...,m−1.ThenifuhattainsitsmaximumonΩ∪∂Ω∗atapoint(ih,jh)∈Ω,thenthefollowinghh00hholds:uisconstantonΩ∪∂Ω∗.hhhHere∂Ω∗:=∂Ω{(0,0),(a,0),(0,b),(a,b)}.hhInparticular,wehavemaxuh(x,y)≤maxuh(x,y).(x,y)∈Ωh(x,y)∈∂Ω∗hProof:Let¯u:=uh(i0h,j0h).Thenbecauseof(1.7)andfij≤0wehave4¯u≤uh(kh,lh)≤4¯u,(k,l)∈N(i0,j0)sinceinparticularuh(kh,lh)≤u¯for(k,l)∈N(i0,j0).HereweusedthenotationN(i0,j0)={((i0−1),j0),((i0+1),j0),(i0,(j0+1)),(i0,(j0−1))}forthesetofindicesofneighboursof(i0h,j0h)inthefive-pointstencil.Fromtheseinequalitiesweconcludethatuh(kh,lh)=¯ufor(k,l)∈N(i0,j0).IfweapplythisargumenttotheneighboursinΩhofthegridpoints(kh,lh)for(k,l)∈N(i0,j0)andthencontinueinthesamewaytothesetsofneigh-boursinΩharisingineverysuchstep,thenfinally,foreachgridpoint(ih,jh)∈Ω∪∂Ω∗theclaimedidentityu(ih,jh)=¯uisachieved.
hhhTheexceptionalsetofvertices∂Ω∂Ω∗doesnotparticipateinanyhhdifferencestencil,sothatthevaluesthereareofnorelevanceforuh.Wewanttogeneralizethisresultandthereforeconsiderasystemofequationsasin(1.10),(1.11):Ahuh=qh=−Aˆhuˆh+f,(1.31)whereA∈RM1,M1asin(1.10),Aˆ∈RM1,M2asin(1.11),u,f∈RM1,hhhanduˆ∈RM2.Thismaybeinterpretedasthediscretizationofabound-haryvalueproblemobtainedbythefinitedifferencemethodoranyotherapproachandwithoutrestrictionsonthedimensionalityofthedomain.AtleastononepartoftheboundaryDirichletboundaryconditionsarere-quired.Thentheentriesofthevectoruhcanbeinterpretedastheunknown(1)(1)valuesatthegridpointsinΩh∪∂Ωh,where∂Ωhcorrespondtoapartof∂Ω(withfluxormixedboundarycondition).Analogously,thevectoruˆh 381.FiniteDifferenceMethodforthePoissonEquation(indexedfromM1+1toM1+M2)correspondstothevaluesfixedbythe(2)Dirichletboundaryconditionson∂Ωh.AgainletM=M1+M2andA˜:=AAˆ∈RM1,M.hhhThismeansinparticularthatthedimensionsM1andM2arenotfixed,butareingeneralunboundedforh→0.Orientedon(1.15)werequirethefollowinggeneralassumptionsfortherestofthesection:(1)(Ah)rr>0forallr=1,...,M1,(2)(Ah)rs≤0forallr,s=1,...,M1suchthatr=s,M1(3)(i)(Ah)rs≥0forallr=1,...,M1,s=1(ii)foratleastoneindexthestrictinequalityholds,(4)Ahisirreducible,(1.32)(5)(Aˆh)rs≤0forallr=1,...,M1,s=M1+1,...,M,M(6)(A˜h)rs≥0forallr=1,...,M1,s=1(7)foreverys=M1+1,...,Mthereexistsr∈{1,...,M1}suchthat(Aˆh)rs=0.Generalizingthenotationaboveforr∈{1,...,M1},theindicess∈{1,...,M}{r}arecalledneighbours,forwhich(A˜h)rs=0,andtheyareassembledtoformthesetNr.Therefore,theirreducibilityofAhmeansthatarbitraryr,s∈{1,...,M1}canbeconnectedbyneighbourhoodrelationships.Thecondition(7)isnotarestriction:Itonlyavoidstheinclusionofknownvalues(uˆh)sthatdonotinfluencethesolutionof(1.31)atall.Forthefive-pointstencilontherectangle,thesearethevaluesatthecornerpoints.Becauseofthecondition(7),everyindexr∈{M1+1,...,M}isconnectedtoeveryindexs∈{1,...,M1}bymeansofneighbourhoodrelationships.Theconditions(2)and(3)implytheweakdiagonaldominanceofAh.Notethattheconditionsareformulatedredundantly:Thecondition(3)alsofollowsfrom(5)through(7).Tosimplifythenotationwewillusethefollowingconventions,whereu,vandA,Barevectorsandmatrices,respectively,ofsuitabledimensions:u≥0ifandonlyif(u)i≥0forallindicesi,u≥vifandonlyifu−v≥0,(1.33)A≥0ifandonlyif(A)ij≥0forallindices(i,j),A≥BifandonlyifA−B≥0. 1.4.MaximumPrinciplesandStability39Theorem1.9Weconsider(1.31)undertheassumptions(1.32).Fur-thermore,letf≤0.Thenastrongmaximumprincipleholds:Iftheucomponentsofu˜=hattainanonnegativemaximumforsomein-huˆhdexr∈{1,...,M1},thenallthecomponentsareequal.Inparticular,aweakmaximumprincipleisfulfilled:max(u˜h)r≤max0,max(uˆh)r.(1.34)r∈{1,...,M}r∈{M1+1,...,M}Proof:Let¯u=maxs∈{1,...,M}(u˜h)s,and¯u=(uh)rwherer∈{1,...,M1}.Becauseof(1.32)(2),(5),(6)therthrowof(1.31)implies(Ah)rru¯≤−A˜hrs(u˜h)s=A˜hrs(u˜h)ss∈Nrs∈Nr(1.35)≤A˜hrsu¯≤(Ah)rru,¯s∈Nrwheretheassumption¯u≥0isusedinthelastestimate.Therefore,ev-erywhereequalityhastohold.Sincethesecondinequalityisvalidalsoforeverysingletermand(A˜h)rs=0bythedefinitionofNr,wefinallyconcludethat(u˜h)s=¯uforalls∈Nr.Thisallowsustoapplythisargumenttoalls∈Nr∩{1,...,M1},thentothecorrespondingsetsofneighbours,andsoon,untiltheassertionisproven.
Therequirementofirreducibilitycanbeweakenedifinsteadof(1.32)(6)wehaveM(6)∗A˜=0forallr=1,...,M.hrs1s=1Thencondition(4)canbereplacedbytherequirement∗(4)Foreveryr1∈{1,...,M1}suchthatM1(Ah)r1s=0(1.36)s=1thereareindicesr2,...,rl+1suchthat(Ah)riri+1=0fori=1,...,landM1(Ah)rl+1s>0.(1.37)s=1∗Thesemodifiedconditionswithout(7)willbedenotedby(1.32). 401.FiniteDifferenceMethodforthePoissonEquationMotivatedbytheexampleabovewecallapointr∈{1,...,M1}farfromtheboundaryif(1.36)holds,andclosetotheboundaryif(1.37)holds,andthepointsr∈{M1+1,...,M}arecalledboundarypoints.Theorem1.10Weconsider(1.31)undertheassumption(1.32)∗.Iff≤0,thenmax(u˜h)r≤max(uˆh)r.(1.38)r∈{1,...,M}r∈{M1+1,...,M}Proof:WeusethesamenotationandthesameargumentsasintheproofofTheorem1.9.In(1.35)inthelastestimateequalityholds,sothatnosignconditionsfor¯uarenecessary.Becauseof(4)∗themaximumwillalsobeattainedatapointclosetotheboundaryandthereforealsoatitsneighbours.Becauseof(6)∗aboundarypointalsobelongstotheseneighbours,whichprovestheassertion.
Fromthemaximumprinciplesweimmediatelyconcludeacomparisonprinciple:Lemma1.11Weassume(1.32)or(1.32)∗.Letu,u∈RM1besolutionsofh1h2Ahuhi=−Aˆhuˆhi+fifori=1,2forgivenf,f∈RM1,uˆ,uˆ∈RM2,whichsatisfyf≤f,uˆ≤12h1h212h1uˆh2.Thenuh1≤uh2.Proof:FromAh(uh1−uh2)=−Aˆh(uˆh1−uˆh2)+f1−f2wecanconcludewithTheorem1.9or1.10thatmax(uh1−uh2)r≤0.r∈{1,...,M1}
Thisimpliesinparticulartheuniquenessofasolutionof(1.31)forarbitraryuˆhandfandalsotheregularityofAh.Inthefollowingwedenoteby0and0thezerovectorandthezeromatrix,respectively,whereallcomponentsareequalto0.AnimmediateconsequenceofLemma1.11isthefollowingTheorem1.12LetA∈RM1,M1beamatrixwiththeproperties(1.32)h(1)–(3)(i),(4)∗,andu∈RM1.ThenhAhuh≥0impliesuh≥0.(1.39)Proof:TobeabletoapplyLemma1.11,onehastoconstructamatrixAˆ∈RM1,M2suchthat(1.32)*holds.Obviously,thisispossible.Thenoneh 1.4.MaximumPrinciplesandStability41canchooseuh2:=uh,f2:=Ahuh2,uˆh2:=0,uh1:=0,f1:=0,uˆh1:=0toconcludetheassertion.Becauseofuˆhi:=0fori=1,2thespecificdefinitionofAˆhplaysnorole.
Amatrixwiththeproperty(1.39)iscalledinversemonotone.Anequivalentrequirementis−1vh≥0⇒Ahvh≥0,andthereforebychoosingtheunitvectorsasvh,−1A≥0.hInversemonotonematricesthatalsosatisfy(1.32)(1),(2)arecalledM-matrices.Finally,wecanweakentheassumptionsforthevalidityofthecomparisonprinciple.Corollary1.13SupposethatA∈RM1,M1isinversemonotoneandh(1.32)(5)holds.Letu,u∈RM1besolutionsofh1h2Ahuhi=−Aˆhuˆhi+fifori=1,2forgivenf,f∈RM1,uˆ,uˆ∈RM2thatsatisfyf≤f,uˆ≤uˆ.12h1h212h1h2Thenuh1≤uh2.Proof:MultiplyingtheequationAh(uh1−uh2)=−Aˆh(uˆh1−uˆh2)+f1−f2−1fromtheleftbythematrixA,wegethu−u=−A−1Aˆ(uˆ−uˆ)+A−1(f−f)≤0.h1h2h!hh1h2!h12!!!≥0≤0≤0≥0≤0
TheimportanceofCorollary1.13liesinthefactthatthereexistdiscretizationmethods,forwhichthematrixA˜hdoesnotsatisfy,e.g.,con-dition(1.32)(6),or(6)∗butA−1≥0.AtypicalexampleofsuchamethodhisthefinitevolumemethoddescribedinChapter6.Inthefollowingwedenoteby1avector(ofsuitabledimension)whosecomponentsareallequalto1.Theorem1.14Weassume(1.32)(1)–(3),(4)∗,(5).Furthermore,letw(1),w(2)∈RM1begivensuchthathh(1)(2)Ahwh≥1,Ahwh≥−Aˆh1.(1.40) 421.FiniteDifferenceMethodforthePoissonEquationThenasolutionofAhuh=−Aˆhuˆh+fsatisfies(1)(2)(1)(2)(1)−|f|∞wh+|uˆh|∞wh≤uh≤|f|∞wh+|uˆh|∞wh,(2)|u|≤w(1)|f|+w(2)|uˆ|.h∞h∞∞h∞h∞Undertheassumptions(1.32)(1)–(3),(4)∗,and(1.40)thematrixnorm·∞inducedby|·|∞satisfiesA−1≤w(1).h∞h∞Proof:Since−|f|∞1≤f≤|f|∞1andtheanalogousstatementforuˆh(1)(2)isvalid,thevectorvh:=|f|∞wh+|uˆh|∞wh−uhsatisfiesAhvh≥|f|∞1−f−Aˆh(|uˆh|∞1−uˆh)≥0,wherewehavealsoused−Aˆh≥0inthelastestimate.Therefore,therightinequalityof(1)impliesfromTheorem1.12thattheleftinequalitycanbeprovenanalogously.Thefurtherassertionsfollowimmediatelyfrom(1).
Becauseoftheinversemonotonicityandfrom(1.32)(5)thevectorspos-(i)tulatedinTheorem1.14havetosatisfyw≥0necessarilyfori=1,2.hThusstabilitywithrespectto·∞ofthemethoddefinedby(1.31)as-suming(1.32)(1)–(3),(4)*isguaranteedifavector0≤w∈RM1andahconstantC>0independentofhcanbefoundsuchthatAhwh≥1and|wh|∞≤C.(1.41)Finally,thiswillbeprovenforthefive-pointstencildiscretization(1.1),(1.2)ontherectangleΩ=(0,a)×(0,b)forC=1(a2+b2).16Forthisreasonwedefinepolynomialsofseconddegreew1,w2by11w1(x):=x(a−x)andw2(y):=y(b−y).(1.42)44Itisclearthatw1(x)≥0forallx∈[0,a]andw2(y)≥0forally∈[0,b].Furthermore,wehavew1(0)=0=w1(a)andw2(0)=0=w2(b),and

1

1w1(x)=−andw2(y)=−.22Thereforewandwarestrictlyconcaveandattaintheirmaximumina122andb,respectively.Thusthefunctionw(x,y):=w(x)+w(x)satisfies212−∆w=1inΩ,(1.43)w≥0on∂Ω.Nowletw∈RM1be,forafixedordering,therepresentationofthegridhfunctionwhdefinedby(wh)(ih,jh):=w(ih,jh)fori=1,...,l−1,j=1,...,m−1. 1.4.MaximumPrinciplesandStability43Analogously,letwˆ∈RM2betherepresentationofthefunctionˆwde-hhfinedon∂Ω∗.AscanbeseenfromtheerrorrepresentationinLemma1.2,hstatement4,thedifferencequotient∂−∂+u(x)isexactforpolynomialsofseconddegree.Therefore,weconcludefrom(1.43)thatAhwh=−Aˆhwˆh+1≥1,whichfinallyimpliesab122|wh|∞=wh∞≤w∞=w1+w2=(a+b).2216Thisexamplemotivatesthefollowinggeneralproceduretoconstructwh∈RM1andaconstantCsuchthat(1.41)isfulfilled.Assumethattheboundaryvalueproblemunderconsiderationreadsinanabstractform(Lu)(x)=f(x)forx∈Ω,(1.44)(Ru)(x)=g(x)forx∈∂Ω.Similarto(1.43)wecanconsider—incaseofexistence—asolutionwof(1.44)forsomef,g,suchthatf(x)≥1forallx∈Ω,g(x)≥0forallx∈Ω.IfwisboundedonΩ,then(wh)i:=w(xi),i=1,...,M1,forthe(non-Dirichlet)gridpointsxi,isacandidateforwh.Obviously,|wh|∞≤w∞.Correspondingly,weset(wˆh)i=w(xi)≥0,i=M1+1,...,M2,fortheDirichlet-boundarygridpoints.Theexactfulfillmentofthediscreteequationsbywhcannotbeexpectedanymore,butincaseofconsistencytheresidualcanbemadearbitrarilysmallforsmallh.ThisleadstoTheorem1.15Assumethatasolutionw∈C(Ω)of(1.44)existsfordataf≥1andg≥0.Ifthediscretizationoftheform(1.31)isconsistentwith(1.44)(forthesedata),andthereexistsH>˜0sothatforsomeα>˜0:−Aˆhwˆh+f≥α˜1forh≤H,˜(1.45)thenforevery0<α<α˜thereexistsH>0,sothatAhwh≥α1forh≤H.Proof:Setτh:=Ahwh+Aˆhwˆh−f 441.FiniteDifferenceMethodforthePoissonEquationfortheconsistencyerror,then|τh|∞→0forh→0.ThusAhwh=τh−Aˆhwˆh+f≥−|τh|∞1+˜α1forh≤H˜≥α1forh≤HandsomeappropriateH>0.
Thusaproperchoicein(1.41)is11whandC:=w∞.(1.46)ααThecondition(1.45)isnotcritical:IncaseofDirichletboundaryconditionsand(1.32)(5)(forcorrespondingrowsiofAˆh)then,dueto(f)i≥1,wecanevenchoose˜α=1.ThediscussionofNeumannboundaryconditionsfollowing(1.24)showsthatthesamecanbeexpected.Theorem1.15showsthatforadiscretizationwithaninversemonotonesystemmatrixconsistencyalreadyimpliesstability.Toconcludethissectionletusdiscussthevariousingredientsof(1.32)or(1.32)*thataresufficientforarangeofpropertiesfromtheinversemonotonicityuptoastrongmaximumprinciple:Forthefive-pointstencilonarectangleallthepropertiesarevalidforDirichletboundaryconditions.IfpartlyNeumannboundaryconditionsappear,thesituationisthesame,butnowcloseandfarfromtheboundaryreferstoitsDirichletpart.IntheinterpretationoftheimplicationsonehastotakeintoaccountthattheheterogeneitiesoftheNeumannboundaryconditionarenowpartoftheright-handsidef,asseen,e.g.,in(1.26).Ifmixedboundaryconditionsareapplied,as∂νu+αu=gonΓ2(1.47)forsomeΓ2⊂Γandα=α(x)>0,thenthesituationisthesameagainifαuisapproximatedjustbyevaluation,atthecostthat(4)*nolongerholds.Thesituationissimilarifreactiontermsappearinthedifferentialequation(seeExercise1.10).Exercises1.11GiveanexampleofamatrixAˆ∈RM1,M2thatcanbeusedinthehproofofTheorem1.12.1.12ShowthatthetranspositionofanM-matrixisagainanM-matrix. Exercises451.13IntheassumptionsofTheorem1.9substitute(1.32)(4)by(4)*andamend(6)to#(6)Condition(1.32)(6)isvalidandM1(Ah)rs>0⇒thereexistss∈{M1,...,M}suchthat(Aˆh)rs<0.s=1UndertheseconditionsproveaweakmaximumprincipleasinTheorem1.9.1.14Assumingtheexistenceofw∈RM1suchthatAw≥1andhhh|wh|∞≤CforsomeconstantCindependentofh,showdirectly(withoutTheorem1.14)arefinedorderofconvergenceestimateonthebasisofanorderofconsistencyestimateinwhichalsotheshapeofwhappears. 2TheFiniteElementMethodforthePoissonEquationThefiniteelementmethod,frequentlyabbreviatedbyFEM,wasdevel-opedinthefiftiesintheaircraftindustry,aftertheconcepthadbeenindependentlyoutlinedbymathematiciansatanearliertime.Eventodaythenotionsusedreflectthatoneoriginofthedevelopmentliesstructuralmechanics.Shortlyafterthisbeginning,thefiniteelementmethodwasap-pliedtoproblemsofheatconductionandfluidmechanics,whichformtheapplicationbackgroundofthisbook.Anintensivemathematicalanalysisandfurtherdevelopmentwasstartedinthelatersixties.Thebasicsofthismathematicaldescriptionandanaly-sisaretobedevelopedinthisandthefollowingchapter.ThehomogeneousDirichletboundaryvalueproblemforthePoissonequationformstheparadigmofthischapter,butmoregenerallyvalidconsiderationswillbeemphasized.InthiswaytheabstractfoundationforthetreatmentofmoregeneralproblemsinChapter3isprovided.Inspiteoftheimportanceofthefiniteelementmethodforstructuralmechanics,thetreatmentofthelinearelasticityequationswillbeomitted.Butwenotethatonlyasmallexpenseisnecessaryfortheapplicationoftheconsiderationstotheseequations.Wereferto[11],wherethisisrealizedwithaverysimilarnotation.2.1VariationalFormulationfortheModelProblemWewilldevelopanewsolutionconceptfortheboundaryvalueproblem(1.1),(1.2)asatheoreticalfoundationforthefiniteelementmethod.For 2.1.VariationalFormulation47suchasolution,thevalidityofthedifferentialequation(1.1)isnolongerre-quiredpointwisebutinthesenseofsomeintegralaveragewith“arbitrary”weightingfunctionsϕ.Inthesameway,theboundarycondition(1.2)willbeweakenedbytherenunciationofitspointwisevalidity.Forthepresent,wewanttoconfinetheconsiderationstothecaseofhomogeneousboundaryconditions(i.e.,g≡0),andsoweconsiderthefollowinghomogeneousDirichletproblemforthePoissonequation:Givenafunctionf:Ω→R,findafunctionu:Ω→Rsuchthat−∆u=finΩ,(2.1)u=0on∂Ω.(2.2)InthefollowingletΩbeadomainsuchthattheintegraltheoremofGaussisvalid,i.e.foranyvectorfieldq:Ω→RdwithcomponentsinC(Ω)∩C1(Ω)itholds

∇·q(x)dx=ν(x)·q(x)dσ.(2.3)Ω∂ΩLetthefunctionu:Ω→Rbeaclassicalsolutionof(2.1),(2.2)inthesenseofDefinition1.1,whichadditionallysatisfiesu∈C1(Ω)tofacili-tatethereasoning.Nextweconsiderarbitraryv∈C∞(Ω)asso-calledtest0functions.Thesmoothnessofthesefunctionsallowsalloperationsofdiffer-entiation,andfurthermore,allderivativesofafunctionv∈C∞(Ω)vanish0ontheboundary∂Ω.Wemultiplyequation(2.1)byv,integratetheresultoverΩ,andobtain

f,v=f(x)v(x)dx=−∇·(∇u)(x)v(x)dx0
ΩΩ
=∇u(x)·∇v(x)dx−∇u(x)·ν(x)v(x)dσ(2.4)
Ω∂Ω=∇u(x)·∇v(x)dx.ΩTheequalitysignatthebeginningofthesecondlineof(2.4)isobtainedbyintegrationbypartsusingtheintegraltheoremofGausswithq=v∇u.Theboundaryintegralvanishesbecausev=0holdson∂Ω.Ifwedefine,foru∈C1(Ω),v∈C∞(Ω),areal-valuedmappingaby0
a(u,v):=∇u(x)·∇v(x)dx,Ωthentheclassicalsolutionoftheboundaryvalueproblemsatisfiestheidentity∞a(u,v)=f,v0forallv∈C0(Ω).(2.5) 482.FiniteElementMethodforPoissonEquationThemappingadefinesascalarproductonC∞(Ω)thatinducesthenorm0"
1/22ua:=a(u,u)=|∇u|dx(2.6)Ω(seeAppendixA.4forthesenotions).Mostofthepropertiesofascalarproductareobvious.Onlythedefiniteness(A4.7)requiresfurtherconsiderations.Namely,wehavetoshowthat
a(u,u)=(∇u·∇u)(x)dx=0⇐⇒u≡0.ΩToprovethisassertion,firstweshowthata(u,u)=0implies∇u(x)=0forallx∈Ω.Todothis,wesupposethatthereexistssomepoint¯x∈Ωsuchthat∇u(¯x)=0.Then(∇u·∇u)(¯x)=|∇u|2(¯x)>0.Becauseofthecontinuityof∇u,asmallneighbourhoodGof¯xexistswithapositivemeasure|G|and|∇u|(x)≥α>0forallx∈G.Since|∇u|2(x)≥0forallx∈Ω,itfollowsthat
22|∇u|(x)dx≥α|G|>0,Ωwhichisincontradictiontoa(u,u)=0.Consequently,∇u(x)=0holdsforallx∈Ω;i.e.,uisconstantinΩ.Sinceu(x)=0forallx∈∂Ω,theassertionfollows.Unfortunately,thespaceC∞(Ω)istoosmalltoplaythepartofthebasic0spacebecausethesolutionudoesnotbelongtoC∞(Ω)ingeneral.The0identity(2.4)istobesatisfiedforalargerclassoffunctions,whichinclude,asanexampleforv,thesolutionuandthefiniteelementapproximationtoutobedefinedlater.ForthepresentwedefineasthebasicspaceV,V:=u:Ω→Ru∈C(Ω)¯,∂iuexistsandispiecewise(2.7)continuousforalli=1,...,d,u=0on∂Ω.Tosaythat∂iuispiecewisecontinuousmeansthatthedomainΩcanbedecomposedasfollows:#Ω=¯Ω¯j,jwithafinitenumberofopensetsΩj,withΩj∩Ωk=∅forj=k,and∂iuiscontinuousonΩjanditcancontinuouslybeextendedonΩ¯j.Thenthefollowingpropertieshold:•aisascalarproductalsoonV,•C∞(Ω)⊂V,0•C∞(Ω)isdenseinVwithrespectto·;i.e.,foranyu∈V(2.8)0aasequence(u)inC∞(Ω)existssuchthatu−u→0nn∈N0naforn→∞, 2.1.VariationalFormulation49•C∞(Ω)isdenseinVwithrespectto·.(2.9)00Thefirstandsecondstatementsareobvious.Thetwoothersrequireacertaintechnicaleffort.AmoregeneralstatementwillbeformulatedinTheorem3.7.Withthat,weobtainfrom(2.5)thefollowingresult:Lemma2.1Letubeaclassicalsolutionof(2.1),(2.2)andletu∈C1(Ω)¯.Thena(u,v)=f,vforallv∈V.(2.10)0Equation(2.10)isalsocalledavariationalequation.Proof:Letv∈V.Thenv∈C∞(Ω)existwithv→vwithrespectn0nto·0andalsoto·a.Therefore,itfollowsfromthecontinuityofthebilinearformwithrespectto·a(see(A4.22))andthecontinuityofthefunctionaldefinedbytheright-handsidev→f,v0withrespectto·0(becauseoftheCauchy–SchwarzinequalityinL2(Ω))thatf,vn0→f,v0anda(u,vn)→a(u,v)forn→∞.Sincea(u,vn)=f,vn0,wegeta(u,v)=f,v0.
ThespaceVintheidentity(2.10)canbefurtherenlargedaslongas(2.8)and(2.9)willremainvalid.Thisfactwillbeusedlatertogiveacorrectdefinition.Definition2.2Afunctionu∈Viscalledaweak(orvariational)solutionof(2.1),(2.2)ifthefollowingvariationalequationholds:a(u,v)=f,vforallv∈V.0Ifumodelse.g.thedisplacementofamembrane,thisrelationiscalledtheprincipleofvirtualwork.Lemma2.1guaranteesthataclassicalsolutionuisaweaksolution.Theweakformulationhasthefollowingproperties:•Itrequireslesssmoothness:∂iuhastobeonlypiecewisecontinuous.•ThevalidityoftheboundaryconditionisguaranteedbythedefinitionofthefunctionspaceV.Wenowshowthatthevariationalequation(2.10)hasexactlythesamesolution(s)asaminimizationproblem:Lemma2.3Thevariationalequation(2.10)hasthesamesolutionsu∈VastheminimizationproblemF(v)→minforallv∈V,(2.11) 502.FiniteElementMethodforPoissonEquationwhere112F(v):=a(v,v)−f,v0=va−f,v0.22Proof:(2.10)⇒(2.11):Letubeasolutionof(2.10)andletv∈Vbechosenarbitrarily.Wedefinew:=v−u∈V(becauseVisavectorspace),i.e.,v=u+w.Then,usingthebilinearityandsymmetry,wehave1F(v)=a(u+w,u+w)−f,u+w2011=a(u,u)+a(u,w)+a(w,w)−f,u−f,w(2.12)22001=F(u)+a(w,w)≥F(u),2wherethelastinequalityfollowsfromthepositivityofa;i.e.,(2.11)holds.(2.10)⇐(2.11):Letubeasolutionof(2.11)andletv∈V,ε∈Rbechosenarbitrarily.Wedefineg(ε):=F(u+εv)forε∈R.Theng(ε)=F(u+εv)≥F(u)=g(0)forallε∈R,becauseu+εv∈V;i.e.,ghasaglobalminimumatε=0.Itfollowsanalogouslyto(2.12):1ε2g(ε)=a(u,u)−f,u+ε(a(u,v)−f,v)+a(v,v).2002Hencethefunctiongisaquadraticpolynomialinε,andinparticular,g∈C1(R)isvalid.Thereforeweobtainthenecessarycondition
0=g(ε)=a(u,v)−f,v0fortheexistenceofaminimumatε=0.Thususolves(2.10),becausev∈Vhasbeenchosenarbitrarily.
Forapplicationse.g.instructuralmechanicsasabove,theminimizationproblemiscalledtheprincipleofminimalpotentialenergy.Remark2.4Lemma2.3holdsforgeneralvectorspacesVifaisasym-metric,positivebilinearformandtheright-handsidef,visreplacedby0b(v),whereb:V→Risalinearmapping,alinearfunctional.Thenthevariationalequationreadsasfindu∈Vwitha(u,v)=b(v)forallv∈V,(2.13)andtheminimizationproblemasfindu∈VwithF(u)=minF(v)v∈V,(2.14) 2.1.VariationalFormulation511whereF(v):=a(v,v)−b(v).2Lemma2.5Theweaksolutionaccordingto(2.10)(or(2.11))isunique.Proof:Letu1,u2betwoweaksolutions,i.e.,a(u1,v)=f,v0,forallv∈V.a(u2,v)=f,v0,Bysubtraction,itfollowsthata(u1−u2,v)=0forallv∈V.Choosingv=u1−u2impliesa(u1−u2,u1−u2)=0andconsequentlyu1=u2,becauseaisdefinite.
Remark2.6Lemma2.5isgenerallyvalidifaisadefinitebilinearformandbisalinearform.Sofar,wehavedefinedtwodifferentnormsonV:·aand·0.ThedifferencebetweenthesenormsisessentialbecausetheyarenotequivalentonthevectorspaceVdefinedby(2.7),andconsequently,theygeneratedifferentconvergenceconcepts,aswillbeshownbythefollowingexample:Example2.7LetΩ=(0,1),i.e.
1a(u,v):=u
v
dx,0andletvn:Ω→Rforn≥2bedefinedby(cf.Figure2.1)nx,for0≤x≤1,nv(x)=1,for1≤x≤1−1,nnnn−nx,for1−1≤x≤1.n1vn1n-1nn1Figure2.1.Thefunctionvn.Then
1/21vn0≤1dx=1,0 522.FiniteElementMethodforPoissonEquation$
1
1%1/2n√v=n2dx+n2dx=2n→∞forn→∞.na01−1nTherefore,thereexistsnoconstantC>0suchthatva≤Cv0forallv∈V.However,aswewillshowinTheorem2.18,thereexistsaconstantC>0suchthattheestimatev0≤Cvaforallv∈Vholds;i.e.,·aisthestrongernorm.ItispossibletoenlargethebasicspaceVwithoutviolatingthepreviousstatements.Theenlargementisalsonecessarybecause,forinstance,theproofoftheexistenceofasolutionofthevariationalequation(2.13)ortheminimizationproblem(2.14)requiresingeneralthecompletenessofV.However,theactualdefinitionofVdoesnotimplythecompleteness,asthefollowingexampleshows:Example2.8LetΩ=(0,1)againandtherefore
1a(u,v):=u
v
dx.0Foru(x):=xα(1−x)αwithα∈1,1weconsiderthesequenceoffunctions2&'u(x)forx∈1,1−1,nn&'u(x):=nu(1)xforx∈0,1,nnn&'nu(1−1)(1−x)forx∈1−1,1.nnThenun−uma→0forn,m→∞,un−ua→0forn→∞,butu/∈V,whereVisdefinedanalogouslyto(2.7)withd=1.InSection3.1wewillseethatavectorspaceV˜normedwith·aexistssuchthatu∈V˜andV⊂V˜.Therefore,Visnotcompletewithrespectto·a;otherwise,u∈Vmustbevalid.Infact,thereexistsa(unique)completionofVwithrespectto·a(seeAppendixA.4,especially(A4.26)),butwehavetodescribethenew“functions”addedbythisprocess.Besides,integrationbypartsmustbevalidsuchthataclassicalsolutioncontinuestobealsoaweaksolution(comparewithLemma2.1).Therefore,thefollowingideaisunsuitable.AttemptofacorrectdefinitionofV:LetVbethesetofalluwiththepropertythat∂iuexistsforallx∈Ωwithoutanyrequirementson∂iuinthesenseofafunction. 2.1.VariationalFormulation53Forinstance,thereexistsCantor’sfunctionwiththefollowingproperties:f:[0,1]→R,f∈C([0,1]),f=0,fisnotconstant,f
(x)existswithf
(x)=0forallx∈[0,1].xHerethefundamentaltheoremofcalculus,f(x)=f
(s)ds+f(0),and0thustheprincipleofintegrationbyparts,arenolongervalid.Consequently,additionalconditionsfor∂iuarenecessary.ToprepareanadequatedefinitionofthespaceV,weextendthedefinitionofderivativesbymeansoftheiractiononaveragingprocedures.Inordertodothis,weintroducethemulti-indexnotation.Avectorα=(α1,...,αd)ofnonnegativeintegersαi∈{0,1,2,...}isdcalledamulti-index.Thenumber|α|:=i=1αidenotestheorder(orlength)ofα.Forx∈Rdletxα:=xα1···xαd.(2.15)1dAshorthandnotationforthedifferentialoperationscanbeadoptedbythis:Foranappropriatelydifferentiablefunctionulet∂αu:=∂α1···∂αdu.(2.16)1dWecanobtainthisdefinitionfrom(2.15)byreplacingxbythesymbolicvectorT∇:=(∂1,...,∂d)ofthefirstpartialderivatives.Forexample,ifd=2andα=(1,2),then|α|=3and∂3u∂αu=∂∂2u=.12∂x∂x212Nowletαbeamulti-indexoflengthkandletu∈Ck(Ω).Wethenobtainforarbitrarytestfunctionsϕ∈C∞(Ω)byintegrationbyparts0

∂αuϕdx=(−1)ku∂αϕdx.ΩΩTheboundaryintegralsvanishbecause∂βϕ=0on∂Ωforallmulti-indicesβ.Therefore,wemakethefollowingdefinition:Definition2.9v∈L2(Ω)iscalledtheweak(orgeneralized)derivative∂αuofu∈L2(Ω)forthemulti-indexαifforallϕ∈C∞(Ω),0

vϕdx=(−1)|α|u∂αϕdx.ΩΩ 542.FiniteElementMethodforPoissonEquationTheweakderivativeiswell-definedbecauseitisunique:Letv1,v2∈L2(Ω)betwoweakderivativesofu.Itfollowsthat
∞(v1−v2)ϕdx=0forallϕ∈C0(Ω).ΩSinceC∞(Ω)isdenseinL2(Ω),wecanfurthermoreconcludethat0
2(v1−v2)ϕdx=0forallϕ∈L(Ω).ΩIfwenowchoosespecificallyϕ=v1−v2,weobtain
v−v2=(v−v)(v−v)dx=0,1201212Ωandv=v(a.e.)followsimmediately.Inparticular,u∈Ck(Ω)hasweak¯12derivatives∂αuforαwith|α|≤k,andtheweakderivativesareidenticaltotheclassical(pointwise)derivatives.AlsothedifferentialoperatorsofvectorcalculuscanbegivenaweakdefinitionanalogoustoDefinition2.9.Forexample,foravectorfieldqwithcomponentsinL2(Ω),v∈L2(Ω)istheweakdivergencev=∇·qifforallϕ∈C∞(Ω)0

vϕdx=−q·∇ϕdx.ΩΩThecorrectchoiceofthespaceVisthespaceH1(Ω),whichwillbe0definedbelow.FirstwedefineH1(Ω):=u:Ω→Ru∈L2(Ω),uhasweakderivatives(2.17)∂u∈L2(Ω)foralli=1,...,d.iAscalarproductonH1(Ω)isdefinedby

u,v:=u(x)v(x)dx+∇u(x)·∇v(x)dx(2.18)1ΩΩwiththenorm(

1/2u:=u,u=|u(x)|2dx+|∇u(x)|2dx(2.19)11ΩΩinducedbythisscalarproduct.Theabove“temporary”definition(2.7)ofVtakescareoftheboundaryconditionu=0on∂Ωbyconditionsforthefunctions.I.e.wewanttochoosethebasicspaceVanalogouslyas:H1(Ω):=u∈H1(Ω)u=0on∂Ω.(2.20)0HereH1(Ω)andH1(Ω)arespecialcasesofso-calledSobolevspaces.0ForΩ⊂Rd,d≥2,H1(Ω)maycontainunboundedfunctions.Inpar-ticular,wehavetoexaminecarefullythemeaningofu|∂Ω(∂Ωhasthe 2.2.TheFiniteElementMethodwithLinearElements55d-dimensionalmeasure0)and,inparticular,u=0on∂Ω.ThiswillbedescribedinSection3.1.Exercises2.1(a)Considertheinterval(−1,1);provethatthefunctionu(x)=|x|hasthegeneralizedderivativeu
(x)=sign(x).(b)Doessign(x)haveageneralizedderivative?)N22.2LetΩ=l=1Ωl,N∈N,wheretheboundedsubdomainsΩl⊂Rarepairwisedisjointandpossesspiecewisesmoothboundaries.Showthatafunctionu∈C(Ω)withu|∈C1(Ω),1≤l≤N,hasaweakderivativeΩll2)N∂iu∈L(Ω),i=1,2,thatcoincidesinl=1Ωlwiththeclassicalone.2.3LetVbethesetoffunctionsthatarecontinuousandpiecewisecon-tinuouslydifferentiableon[0,1]andthatsatisfytheadditionalconditionsu(0)=u(1)=0.ShowthatthereexistinfinitelymanyelementsinVthatminimizethefunctional
12F(u):=1−[u
(x)]2dx.02.2TheFiniteElementMethodwithLinearElementsTheweakformulationoftheboundaryvalueproblem(2.1),(2.2)leadstoparticularcasesofthefollowinggeneral,hereequivalent,problems:LetVbeavectorspace,leta:V×V→Rbeabilinearform,andletb:V→Rbealinearform.Variationalequation:Findu∈Vwitha(u,v)=b(v)forallv∈V.(2.21)Minimizationproblem:Findu∈VwithF(u)=minF(v)v∈V,1(2.22)whereF(v)=a(v,v)−b(v).2Thediscretizationapproachconsistsinthefollowingprocedure:ReplaceVbyafinite-dimensionalsubspaceVh;i.e.,solveinsteadof(2.21)thefinite-dimensionalvariationalequation,finduh∈Vhwitha(uh,v)=b(v)forallv∈Vh.(2.23) 562.FiniteElementMethodforPoissonEquationThisapproachiscalledtheGalerkinmethod.Orsolveinsteadof(2.22)thefinite-dimensionalminimizationproblem,finduh∈VhwithF(uh)=minF(v)v∈Vh.(2.24)ThisapproachiscalledtheRitzmethod.ItisclearfromLemma2.3andRemark2.4thattheGalerkinmethodandtheRitzmethodareequivalentforapositiveandsymmetricbilinearform.Thefinite-dimensionalsubspaceVhiscalledanansatzspace.ThefiniteelementmethodcanbeinterpretedasaGalerkinmethod(andinourexampleasaRitzmethod,too)foranansatzspacewithspecialproperties.Inthefollowing,thesepropertieswillbeextractedbymeansofthesimplestexample.LetVbedefinedby(2.7)orletV=H1(Ω).0Theweakformulationoftheboundaryvalueproblem(2.1),(2.2)correspondstothechoice

a(u,v):=∇u·∇vdx,b(v):=fvdx.ΩΩLetΩ⊂R2beadomainwithapolygonalboundary;i.e.,theboundaryΓofΩconsistsofafinitenumberofstraight-linesegmentsasshowninFigure2.2.ΩFigure2.2.Domainwithapolygonalboundary.LetThbeapartitionofΩintoclosedtrianglesK(i.e.,includingtheboundary∂K)withthefollowingproperties:(1)Ω=∪K∈ThK;(2)ForK,K
∈T,K=K
,h
int(K)∩int(K)=∅,(2.25)whereint(K)denotestheopentriangle(withouttheboundary∂K).(3)IfK=K
butK∩K
=∅,thenK∩K
iseitherapointoracommonedgeofKandK
(cf.Figure2.3).ApartitionofΩwiththeproperties(1),(2)iscalledatriangulationofΩ.If,inaddition,apartitionofΩsatisfiesproperty(3),itiscalledaconformingtriangulation(cf.Figure2.4). 2.2.LinearElements57notallowed:allowed:Figure2.3.Triangulations.ThetrianglesofatriangulationwillbenumberedK1,...,KN.Thesubscripthindicatesthefinenessofthetriangulation,e.g.,h:=maxdiam(K)K∈Th,wherediam(K):=sup|x−y|x,y∈KdenotesthediameterofK.Thusherehisthemaximumlengthoftheedgesofallthetriangles.Sometimes,K∈Thisalsocalleda(geometric)elementofthepartition.Theverticesofthetrianglesarecalledthenodes,andtheywillbenumbereda1,a2,...,aM,i.e.,ai=(xi,yi),i=1,...,M,whereM=M1+M2anda1,...,aM1∈Ω,(2.26)aM1+1,...,aM∈∂Ω.Thiskindofarrangementofthenodesischosenonlyforthesakeofsimplicityofthenotationandisnotessentialforthefollowingconsiderations.a10a8••a•9•K1K2K5a11
a1K•K3K46•
Kaa7
12••2

K10a3a•K11K8K47•K9a5•a6Figure2.4.AconformingtriangulationwithN=12,M=11,M1=3,M2=8.Anapproximationoftheboundaryvalueproblem(2.1),(2.2)withlinearfiniteelementsonagiventriangulationThofΩisobtainediftheansatzspaceVhisdefinedasfollows:Vh:=u∈C(Ω)¯u|K∈P1(K)forallK∈Th,u=0on∂Ω.(2.27) 582.FiniteElementMethodforPoissonEquationHereP1(K)denotesthesetofpolynomialsoffirstdegree(in2variables)onK;i.e.,p∈P1(K)⇔p(x,y)=α+βx+γyforall(x,y)∈Kandforfixedα,β,γ∈R.Sincep∈P1(K)isalsodefinedonthespaceR×R,weusetheshortbutinaccuratenotationP1=P1(K);accordingtothecontext,thedomainofdefinitionwillbegivenasR×Rorasasubsetofit.WehaveVh⊂V.ThisisclearforthecaseofdefinitionofVby(2.7)because∂xu|K=const,∂u|=constforK∈Tforallu∈V.IfV=H1(Ω),thenthisinclusionyKhh0isnotsoobvious.AproofwillbegiveninTheorem3.20below.Anelementu∈Vhisdetermineduniquelybythevaluesu(ai),i=1,...,M1(thenodalvalues).Inparticular,thegivennodalvaluesalreadyenforcethecontinuityofthepiecewiselinearcomposedfunctions.Correspondingly,thehomogeneousDirichletboundaryconditionissatisfiedifthenodalvaluesattheboundarynodesaresettozero.Inthefollowing,wewilldemonstratethesepropertiesbyanunnecessar-ilyinvolvedproof.ThereasonisthatthisproofwillintroducealloftheconsiderationsthatwillleadtoanalogousstatementsforthemoregeneralproblemsofSection3.4.LetXhbethelargeransatzspaceconsistingofcontinuous,piecewiselinearfunctionsbutregardlessofanyboundaryconditions,i.e.,Xh:=u∈C(Ω)¯u|K∈P1(K)forallK∈Th.Lemma2.10Forgivenvaluesatthenodesa1,...,aM,theinterpolationprobleminXhisuniquelysolvable.Thatis,ifthevaluesu1,...,uMaregiven,thenthereexistsauniquelydeterminedelementu∈Xhsuchthatu(ai)=ui,i=1,...,M.Ifuj=0forj=M1+1,...,M,thenitiseventruethatu∈Vh.Proof:(1)ForanyarbitraryK∈Thweconsiderthelocalinterpolationproblem:Findp=pK∈P1suchthatp(ai)=ui,i=1,2,3,(2.28)whereai,i=1,2,3,denotetheverticesofK,andthevaluesui,i=1,2,3,aregiven.Firstweshowthatproblem(2.28)isuniquelysolvableforaparticulartriangle.Asolutionof(2.28)fortheso-calledreferenceelementKˆ(cf.Figure2.5)withtheverticesˆa1=(0,0),ˆa2=(1,0),ˆa3=(0,1)isgivenbyp(x,y)=u1N1(x,y)+u2N2(x,y)+u3N3(x,y) 2.2.LinearElements59y1^K01xFigure2.5.ReferenceelementKˆ.withtheshapefunctionsN1(x,y)=1−x−y,N2(x,y)=x,(2.29)N3(x,y)=y.Evidently,Ni∈P1,andfurthermore,1fori=j,Ni(ˆaj)=δij=fori,j=1,2,3,0fori=j,andthus3p(ˆaj)=uiNi(ˆaj)=ujforallj=1,2,3.i=1Theuniquenessofthesolutioncanbeseeninthefollowingway:Ifp1,p2satisfytheinterpolationproblem(2.28)forthereferenceelement,thenforp:=p1−p2∈P1wehavep(ˆai)=0,i=1,2,3.Herepisgivenintheformp(x,y)=α+βx+γy.Ifwefixthesecondvariabley=0,weobtainapolynomialfunctionofonevariablep(x,0)=α+βx=:q(x)∈P1(R).Thepolynomialqsatisfiesq(0)=0=q(1),andq≡0followsbytheuniquenessofthepolynomialinterpolationinonevariable;i.e.,α=β=0.Analogously,weconsiderq(y):=p(0,y)=α+γy=γy,andweobtainfromq(1)=0thatγ=0andconsequentlyp≡0.Infact,thisadditionalproofofuniquenessisnotnecessary,becausetheuniquenessalreadyfollowsfromthesolvabilityoftheinterpolationproblembecauseofdimP1=3(comparewithSection3.3).NowweturntothecaseofageneraltriangleK.AgeneraltriangleKismappedontoKˆbyanaffinetransformation(cf.Figure2.6)F:Kˆ→K,F(ˆx)=Bxˆ+d,(2.30)whereB∈R2,2,d∈R2aresuchthatF(ˆa)=a.ii 602.FiniteElementMethodforPoissonEquationB=(b1,b2)anddaredeterminedbytheverticesaiofKasfollows:a1=F(ˆa1)=F(0)=d,a2=F(ˆa2)=b1+d=b1+a1,a3=F(ˆa3)=b2+d=b2+a1;i.e.,b1=a2−a1andb2=a3−a1.ThematrixBisregularbecausea2−a1anda3−a1arelinearlyindependent,ensuringF(ˆai)=ai.Since$%33K=conv{a1,a2,a3}:=λiai0≤λi≤1,λi=1i=1i=1andespeciallyKˆ=conv{aˆ1,aˆ2,aˆ3},F[Kˆ]=Kfollowsfromthefactthattheaffine-linearmappingFsatisfies333Fλiaˆi=λiF(ˆai)=λiaii=1i=1i=13for0≤λi≤1,i=1λi=1.Inparticular,theedges(whereoneλiisequalto0)ofKˆaremappedontotheedgesofK.ya3y^1Ka2a1^K01xx^Figure2.6.Affine-lineartransformation.Analogously,theconsiderationscanbeappliedtothespaceRdwordforwordbyreplacingthesetofindices{1,2,3}by{1,...,d+1}.ThiswillbedoneinSection3.3.ThepolynomialspaceP1doesnotchangeundertheaffinetransforma-tionF.(2)Wenowprovethatthelocalfunctionsu|Kcanbecomposedcontinuously:ForeveryK∈Th,letpK∈P1betheuniquesolutionof(2.28),wherethevaluesu1,u2,u3arethevaluesui1,ui2,ui3(i1,i2,i3∈{1,...,M})thathavetobeinterpolatedatthesenodes.LetK,K
∈TbetwodifferentelementsthathaveacommonedgeE.hThenpK=pKonEistobeshown.ThisisvalidbecauseEcanbemappedonto[0,1]×{0}byanaffinetransformation(cf.Figure2.7).Then 2.2.LinearElements61q1(x)=pK(x,0)andq2(x):=pK(x,0)areelementsofP1(R),andtheysolvethesameinterpolationproblematthepointsx=0andx=1;thusq1≡q2.01EFigure2.7.Affine-lineartransformationofEonthereferenceelement[0,1].Therefore,thedefinitionofubymeansofu(x)=pK(x)forx∈K∈Th(2.31)isunique,andthisfunctionsatisfiesu∈C(Ω)and¯u∈Xh.(3)Finally,wewillshowthatu=0on∂Ωforudefinedby(2.31)ifui=0(i=M1+1,...,M)fortheboundarynodes.Theboundary∂ΩconsistsofedgesofelementsK∈Th.LetEbesuchanedge;i.e.,Ehastheverticesai1,ai2withij∈{M1+1,...,M}.Thegivenboundaryvaluesyieldu(aij)=0forj=1,2.Bymeansofanaffinetransformationanalogouslytotheaboveoneweobtainthatu|Eisapoly-nomialoffirstdegreeinonevariableandthatu|Evanishesattwopoints.Sou|E=0,andtheassertionfollows.
Thefollowingstatementisanimportantconsequenceoftheuniquesolv-abilityoftheinterpolationprobleminXhirrespectiveofitsparticulardefinition:Theinterpolationconditionsϕi(aj)=δij,j=1,...,M,(2.32)uniquelydeterminefunctionsϕi∈Xhfori=1,...,M.Foranyu∈Xh,wehaveMu(x)=u(ai)ϕi(x)forx∈Ω,(2.33)i=1becauseboththeleft-handsideandtheright-handsidefunctionsbelongtoXhandareequaltou(ai)atx=ai.MTherepresentationu=i=1αiϕiisunique,too,forotherwise,afunc-tionw∈Xh,w=0,suchthatw(ai)=0foralli=1,...,Mwouldexist.Thus{ϕ1,...,ϕM}isabasisofXh,especiallydimXh=M.Thisbasisiscalledanodalbasisbecauseof(2.33).Fortheparticularcaseofapiecewiselinearansatzspaceontriangles,thebasisfunctionsarecalled 622.FiniteElementMethodforPoissonEquationpyramidalfunctionsbecauseoftheirshape.Ifthesetofindicesisre-strictedto{1,...,M1};i.e.,weomitthebasisfunctionscorrespondingtotheboundarynodes,thenabasisofVhwillbeobtainedanddimVh=M1.Summary:Thefunctionvaluesu(ai)atthenodesa1,...,aMarethede-greesoffreedomofu∈Xh,andthevaluesattheinteriorpointsa1,...,aM1arethedegreesoffreedomofu∈Vh.ThefollowingconsiderationisvalidforanarbitraryansatzspaceVhwithabasis{ϕ1,...,ϕM}.TheGalerkinmethod(2.23)readsasfollows:FindMuh=i=1ξiϕi∈Vhsuchthata(uh,v)=b(v)forallv∈Vh.SinceMv=i=1ηiϕiforηi∈R,thisisequivalenttoa(uh,ϕi)=b(ϕi)foralli=1,...,M⇐⇒Maξjϕj,ϕi=b(ϕi)foralli=1,...,M⇐⇒j=1Ma(ϕj,ϕi)ξj=b(ϕi)foralli=1,...,M⇐⇒j=1Ahξ=qh(2.34)withA=(a(ϕ,ϕ))∈RM,M,ξ=(ξ,...,ξ)Tandq=(b(ϕ)).hjiij1MhiiTherefore,theGalerkinmethodisequivalenttothesystemofequations(2.34).Theconsiderationsforderiving(2.34)showthat,inthecaseofequiva-lenceoftheGalerkinmethodwiththeRitzmethod,thesystemofequations(2.34)isequivalenttotheminimizationproblemF(ξ)=minF(η)η∈RM,(2.35)hhwhere1TTFh(η)=ηAhη−qhη.2Becauseofthesymmetryandpositivedefiniteness,theequivalenceof(2.34)and(2.35)canbeeasilyproven,anditformsthebasisfortheCGmethodsthatwillbediscussedinSection5.2.Usually,Ahiscalledstiffnessmatrix,andqhiscalledtheloadvector.Thesenamesoriginatedfrommechanics.Forourmodelproblem,wehave
(Ah)ij=a(ϕj,ϕi)=∇ϕj·∇ϕidx,
Ω(qh)i=b(ϕi)=fϕidx.ΩByapplyingthefiniteelementmethod,wethushavetoperformthefollowingsteps:(1)DeterminationofAh,qh.Thisstepiscalledassembling. 2.2.LinearElements63(2)SolutionofAhξ=qh.Ifthebasisfunctionsϕihavethepropertyϕi(aj)=δij,thenthesolutionofsystem(2.34)satisfiestherelationξi=uh(ai),i.e.,weobtainthevectorofthenodalvaluesofthefiniteelementapproximation.Usingonlythepropertiesofthebilinearforma,weobtainthefollowingpropertiesofAh:•Ahissymmetricforanarbitrarybasis{ϕi}becauseaissymmetric.•Ahispositivedefiniteforanarbitrarybasis{ϕi}becauseforu=Mi=1ξiϕi,TMMMξAhξ=i,j=1ξja(ϕj,ϕi)ξi=j=1ξjaϕj,i=1ξiϕiMM=aj=1ξjϕj,i=1ξiϕi=a(u,u)>0(2.36)forξ=0andthereforeu≡0.Herewehaveusedonlythepositivedefinitenessofa.Thuswehaveproventhefollowinglemma.Lemma2.11TheGalerkinmethod(2.23)hasauniquesolutionifaisasymmetric,positivedefinitebilinearformandifbisalinearform.Infact,aswewillseeinTheorem3.1,thesymmetryofaisnotnecessary.•Foraspecialbasis(i.e.,foraspecificfiniteelementmethod),Ahisasparsematrix,i.e.,onlyafewentries(Ah)ijdonotvanish.Evidently,
(Ah)ij=0⇔∇ϕj·∇ϕidx=0.ΩThiscanhappenonlyifsuppϕi∩suppϕj=∅,asthispropertyisagainnecessaryforsupp∇ϕi∩supp∇ϕj=∅becauseof(supp∇ϕi∩supp∇ϕj)⊂(suppϕi∩suppϕj).Thebasisfunctionϕivanishesonanelementthatdoesnotcontainthenodeaibecauseoftheuniquenessofthesolutionofthelocalinterpolationproblem.Therefore,#suppϕi=K,K∈Thai∈Kcf.Figure(2.8),andthus(Ah)ij=0⇒ai,aj∈KforsomeK∈Th;(2.37)i.e.,ai,ajareneighbouringnodes. 642.FiniteElementMethodforPoissonEquationIfweusethepiecewiselinearansatzspaceontrianglesandifaiisaninteriornodeinwhichLelementsmeet,thenthereexistatmostLnondiagonalentriesintheithrowofAh.Thisnumberisdeterminedonlybythetypeofthetriangulation,anditisindependentofthefinenessh,i.e.,ofthenumberofunknownsofthesystemofequations.suppϕiaiFigure2.8.Supportofthenodalbasisfunction.Example2.12Weconsideragaintheboundaryvalueproblem(2.1),(2.2)onΩ=(0,a)×(0,b)again,i.e.−∆u=finΩ,u=0on∂Ω,underthecondition(1.4).ThetriangulationonwhichthemethodisbasediscreatedbyapartitionofΩintosquareswithedgesoflengthhandbyasubsequentuniformdivisionofeachsquareintotwotrianglesaccordingtoafixedrule(Friedrichs–Kellertriangulation).Inordertodothis,twopossibilities(a)and(b)(seeFigures2.9and2.10)exist.(a)(b)Figure2.9.PossibilitiesofFriedrichs–Kellertriangulation.Inbothcases,anodeaZbelongstosixelements,andconsequently,ithasatmostsixneighbours: 2.2.LinearElements65aNWaNfor(a):for(b):IIIIIIaWaZaEIVVIVaSaSEFigure2.10.Supportofthebasisfunction.Case(a)becomescase(b)bythetransformationx→a−x,y→y.Thistransformationleavesthedifferentialequationortheweakformula-tion,respectively,unchanged.ThustheGalerkinmethodwiththeansatzspaceVhaccordingto(2.27)doesnotchange,becauseP1isinvariantwithrespecttotheabovetransformation.Therefore,thediscretizationmatricesAhaccordingto(2.34)areseentobeidenticalbytakingintoaccounttherenumberingofthenodesbythetransformation.Thusitissufficienttoconsideronlyonecase,say(b).Anodewhichisfarawayfromtheboundaryhas6neighbouringnodesin{a1,...,aM1},anodeclosetotheboundaryhasless.TheentriesofthematrixintherowcorrespondingtoaZdependonthederivativesofthebasisfunctionϕZaswellasonthederivativesofthebasisfunctionscorrespondingtotheneighbouringnodes.ThevaluesofthepartialderivativesofϕZinelementshavingthecommonvertexaZarelistedinTable2.1,wheretheseelementsarenumberedaccordingtoFigure2.10.IIIIIIIVVVI∂ϕ−10110−11Zhhhh∂ϕ−1−101102ZhhhhTable2.1.Derivativesofthebasisfunctions.ThusfortheentriesofthematrixintherowcorrespondingtoaZwehave

&'222(Ah)Z,Z=a(ϕZ,ϕZ)=|∇ϕZ|dx=2(∂1ϕZ)+(∂2ϕZ)dx,I∪...∪VII∪II∪IIIbecausetheintegrandsareequalonIandIV,onIIandV,andonIIIandVI.Therefore

(A)=2(∂ϕ)2dx+2(∂ϕ)2dx=2h−2h2+2h−2h2=4,hZ,Z1Z2ZI∪IIII∪II
(Ah)Z,N=a(ϕN,ϕZ)=∇ϕN·∇ϕZdxI∪II

−1−1=∂2ϕN∂2ϕZdx=−hhdx=−1,I∪III∪II 662.FiniteElementMethodforPoissonEquationbecause∂1ϕZ=0onIIand∂1ϕN=0onI.TheelementIforϕNcorre-spondstotheelementVforϕZ;i.e.,∂1ϕN=0onI,analogously,itfollowsthat∂ϕ=h−1onI∪II.Inthesamewayweget2N(Ah)Z,E=(Ah)Z,W=(Ah)Z,S=−1aswellas
(Ah)Z,NW=a(ϕNW,ϕZ)=∂1ϕNW∂1ϕZ+∂2ϕNW∂2ϕZdx=0.II∪IIIThelastidentityisdueto∂1ϕNW=0onIIIand∂2ϕNW=0onIII,becausetheelementsVandVIforϕZagreewiththeelementsIIIandIIforϕNW,respectively.Analogously,weobtainfortheremainingvalue(Ah)Z,SE=0,suchthatonly5(insteadofthemaximum7)nonzeroentriesperrowexist.Thewayofassemblingthestiffnessmatrixdescribedaboveiscallednode-basedassembling.However,mostofthecomputerprogramsimplementingthefiniteelementmethoduseanelement-basedassembling,whichwillbeconsideredinSection2.4.Ifthenodesarenumberedrowwiseanalogouslyto(1.13)andiftheequa-tionsaredividedbyh2,thenh−2Acoincideswiththediscretizationmatrixh(1.14),whichisknownfromthefinitedifferencemethod.Butheretheright-handsideisgivenby

h−2(q)=h−2fϕdx=h−2fϕdxhiiiΩI∪...∪VIforaZ=aiandthusitisnotidenticaltof(ai),theright-handsideofthefinitedifferencemethod.However,ifthetrapezoidalrule,whichisexactforg∈P1,isappliedtoapproximatetheright-handsideaccordingto
31g(x)dx≈vol(K)g(ai)(2.38)K3i=1foratriangleKwiththeverticesai,i=1,2,3andwiththeareavol(K),then
11212fϕidx≈h(f(aZ)·1+f(aO)·0+f(aN)·0)=hf(aZ).I326Analogousresultsareobtainedfortheothertriangles,andthus
h−2fϕdx≈f(a).iZI∪...∪VIInsummary,wehavethefollowingresult. 2.2.LinearElements67Lemma2.13ThefiniteelementmethodwithlinearfiniteelementsonatriangulationaccordingtoFigure2.9andwiththetrapezoidalruletoap-proximatetheright-handsideyieldsthesamediscretizationasthefinitedifferencemethodfrom(1.7),(1.8).Wenowreturntothegeneralformulation(2.21)–(2.24).TheapproachoftheRitzmethod(2.24),insteadoftheGalerkinmethod(2.23),yieldsanidenticalapproximationbecauseofthefollowingresult.Lemma2.14Ifaisasymmetricandpositivebilinearformandbisalinearform,thentheGalerkinmethod(2.23)andtheRitzmethod(2.24)haveidenticalsolutions.Proof:ApplyLemma2.3withVhinsteadofV.
HencethefiniteelementmethodistheGalerkinmethod(andinourproblemtheRitzmethod,too)foranansatzspaceVhwiththefollowingproperties:•Thecoefficientshavealocalinterpretation(hereasnodalvalues).Thebasisfunctionshaveasmallsupportsuchthat:•thediscretizationmatrixissparse,•theentriesofthematrixcanbeassembledlocally.Finally,fortheboundaryvalueproblem(2.1),(2.2)withthecorrespond-ingweakformulation,weconsiderotheransatzspaces,whichtosomeextentdonothavetheseproperties:(1)InSection3.2.1,(3.28),wewillshowthatmixedboundaryconditionsneednotbeincludedintheansatzspace.Thenwecanchoosethefi-nitedimensionalpolynomialspaceV=span1,x,y,xy,x2,y2,...hforit.Butinthiscase,Ahisadensematrixandill-conditioned.SuchansatzspacesyieldtheclassicalRitz–Galerkinmethods.(2)LetVh=span{ϕ1,...,ϕN}andletϕi≡0satisfy,forsomeλi,a(ϕi,v)=λiϕi,v0forallv∈V,i.e.,theweakformulationoftheeigenvalueproblem−∆u=λuinΩ,u=0on∂Ω,forwhicheigenvalues0<λ1≤λ2≤...andcorrespondingeigen-functionsϕiexistsuchthatϕi,ϕj0=δij(e.g.,see[12,p.335]).ForspecialdomainsΩ,(λi,ϕi)canbedeterminedexplicitly,and(Ah)ij=a(ϕj,ϕi)=λjϕj,ϕi0=λjδij 682.FiniteElementMethodforPoissonEquationisobtained.ThusAhisadiagonalmatrix,andthesystemofequationsAhξ=qhcanbesolvedwithouttoogreatexpense.Butthiskindofassemblingispossiblewithacceptablecostsforspecialcasesonly.(3)The(spectral)collocationmethodconsistsintherequirementthattheequations(2.1),(2.2)besatisfiedonlyatcertaindistinctpointsxi∈Ω,calledcollocationpoints,foraspecialpolynomialspaceVh.TheaboveexamplesdescribeGalerkinmethodswithouthavingthetypicalpropertiesofafiniteelementmethod.2.3StabilityandConvergenceoftheFiniteElementMethodWeconsiderthegeneralcaseofavariationalequationoftheform(2.21)andtheGalerkinmethod(2.23).Hereletabeabilinearform,whichisnotnecessarilysymmetric,andletbbealinearform.Then,ife:=u−uh(∈V)denotestheerror,theimportanterrorequationa(e,v)=0forallv∈Vh(2.39)issatisfied.Toobtainthisequation,itissufficienttoconsiderequation(2.21)onlyforv∈Vh⊂VandthentosubtractfromtheresulttheGalerkinequation(2.23).If,inaddition,aissymmetricandpositivedefinite,i.e.,a(u,v)=a(v,u),a(u,u)≥0,a(u,u)=0⇔u=0(i.e.,aisascalarproduct),thentheerrorisorthogonaltothespaceVhwithrespecttothescalarproducta.Therefore,therelation(2.39)isoftencalledtheorthogonalityoftheerror(totheansatzspace).Ingeneral,theelementuh∈Vhwithminimaldistancetou∈Vwithrespecttotheinducednorm·aischaracterizedby(2.39):Lemma2.15LetVh⊂Vbeasubspace,letabeascalarproductonV,andletu:=a(u,u)1/2bethenorminducedbya.Thenforu∈V,itahhfollowsthata(u−uh,v)=0forallv∈Vh⇔(2.40)u−uha=minu−vav∈Vh.(2.41)Proof:Forarbitrarybutfixedu∈V,letb(v):=a(u,v)forv∈Vh.ThenbisalinearformonVh,so(2.40)isavariationalformulationonVh. 2.3.StabilityandConvergence69AccordingtoLemma2.14orLemma2.3,thisvariationalformulationhasthesamesolutionsasF(uh)=minF(v)v∈Vh11withF(v):=a(v,v)−b(v)=a(v,v)−a(u,v).22Furthermore,Fhasthesameminimaasthefunctional1/21/22F(v)+a(u,u)=a(v,v)−2a(u,v)+a(u,u)1/2=a(u−v,u−v)=u−va,becausetheadditionalterma(u,u)isaconstant.Therefore,Fhasthesameminimaas(2.41).
IfanapproximationuhofuistobesoughtexclusivelyinVh,thentheelementuh,determinedbytheGalerkinmethod,istheoptimalchoicewithrespectto·a.Ageneral,notnecessarilysymmetric,bilinearformaisassumedtosatisfythefollowingconditions,where·denotesanormonV:•aiscontinuouswithrespectto·;i.e.,thereexistsM>0suchthat|a(u,v)|≤Muvforallu,v∈V;(2.42)•aisV-elliptic;i.e.,thereexistsα>0suchthata(u,u)≥αu2foru∈V.(2.43)Ifaisascalarproduct,then(2.42)withM=1and(2.43)(asequality)withα=1arevalidfortheinducednorm·:=·aduetotheCauchy–Schwarzinequality.TheV-ellipticityisanessentialconditionfortheuniqueexistenceofasolutionofthevariationalequation(2.21)andoftheboundaryvalueprob-lemdescribedbyit,whichwillbepresentedinmoredetailinSections3.1and3.2.Italsoimplies—withoutfurtherconditions—thestabilityoftheGalerkinapproximation.Lemma2.16TheGalerkinsolutionuhaccordingto(2.23)isstableinthefollowingsense:1uh≤bindependentlyofh,(2.44)αwhere|b(v)|b:=supv∈V,v=0.v 702.FiniteElementMethodforPoissonEquationProof:Inthecaseuh=0,thereisnothingtoprove.Otherwise,froma(uh,v)=b(v)forallv∈Vh,itfollowsthat2|b(uh)|αuh≤a(uh,uh)=b(uh)≤uh≤buh.uhDividingthisrelationbyαuh,wegettheassertion.
Moreover,theapproximationproperty(2.41)holdsuptoaconstant:Theorem2.17(C´ea’slemma)Assume(2.42),(2.43).ThenthefollowingerrorestimatefortheGalerkinsolutionholds:Mu−uh≤minu−vv∈Vh.(2.45)αProof:Ifu−uh=0,thenthereisnothingtoprove.Otherwise,letv∈Vhbearbitrary.Becauseoftheerrorequation(2.39)anduh−v∈Vh,a(u−uh,uh−v)=0.Therefore,using(2.43)wehaveαu−u2≤a(u−u,u−u)=a(u−u,u−u)+a(u−u,u−v)hhhhhhh=a(u−uh,u−v).Furthermore,bymeansof(2.42)weobtain2αu−uh≤a(u−uh,u−v)≤Mu−uhu−vforarbitraryv∈Vh.Thustheassertionfollowsbydivisionbyαu−uh.
Thereforealsoingeneral,inordertogetanasymptoticerrorestimateinh,itissufficienttoestimatethebestapproximationerrorofVh,i.e.,minu−vv∈Vh.However,thisconsiderationismeaningfulonlyinthosecaseswhereM/αisnottoolarge.Section3.2showsthatthisconditionisnolongersatisfiedforconvection-dominatedproblems.Therefore,theGalerkinapproachhastobemodified,whichwillbedescribedinChapter9.Wewanttoapplythetheorydevelopeduptonowtotheweakformula-tionoftheboundaryvalueproblem(2.1),(2.2)withVaccordingto(2.7)or(2.20)andVhaccordingto(2.27).Accordingto(2.4)thebilinearformaandthelinearformbreadas

a(u,v)=∇u·∇vdx,b(v)=fvdx.ΩΩInordertoguaranteethatthelinearformbiswell-definedonV,itissuffi-cienttoassumethattheright-handsidefoftheboundaryvalueproblembelongstoL2(Ω). 2.3.StabilityandConvergence71SinceaisascalarproductonV,
1/2u=u=|∇u|2dxaΩisanappropriatenorm.Alternatively,thenormintroducedin(2.19)forV=H1(Ω)canbetakenas0

1/2u=|u(x)|2dx+|∇u(x)|2dx.1ΩΩInthelattercase,thequestionariseswhethertheconditions(2.42)and(2.43)arestillsatisfied.Indeed,|a(u,v)|≤uava≤u1v1forallu,v∈V.ThefirstinequalityfollowsfromtheCauchy–Schwarzinequalityforthescalarproducta,andthesecondinequalityfollowsfromthetrivialestimate
1/2u=|∇u(x)|2dx≤uforallu∈V.a1ΩThusaiscontinuouswithrespectto·1withM=1.TheV-ellipticityofa,i.e.,theproperty22a(u,u)=ua≥αu1forsomeα>0andallu∈V,isnotvalidingeneralforV=H1(Ω).However,inthepresentsituationofV=H1(Ω)itisvalidbecauseoftheincorporationoftheboundary0conditionintothedefinitionofV:Theorem2.18(Poincar´e)LetΩ⊂Rnbeopenandbounded.ThenaconstantC>0exists(dependingonΩ)suchthat
1/2u≤C|∇u(x)|2dxforallu∈H1(Ω).00ΩProof:Cf.[13].Foraspecialcase,seeExercise2.5.
Thus(2.43)issatisfied,forinstancewith1α=,1+C2(seealso(3.26)below)andthusinparticular222αu1≤a(u,u)=ua≤u1forallu∈V,(2.46) 722.FiniteElementMethodforPoissonEquationi.e.,thenorms·and·areequivalentonV=H1(Ω)andtherefore1a0theygeneratethesameconvergenceconcept:uh→uwithrespectto·1⇔uh−u1→0⇔uh−ua→0⇔uh→uwithrespectto·a.Insummarytheestimate(2.45)holdsfor·=·1withtheconstant1/α.BecauseoftheCauchy–SchwarzinequalityforthescalarproductonL2(Ω)and
b(v)=f(x)v(x)dx,Ωi.e.,|b(v)|≤f0v0≤f0v1,andthusb≤f0,thestabilityestimate(2.44)foraright-handsidef∈L2(Ω)takestheparticularform1uh1≤f0.αUptonow,ourconsiderationshavebeenindependentofthespecialformofVh.NowwemakeuseofthechoiceofVhaccordingto(2.27).InordertoobtainanestimateoftheapproximationerrorofVh,itissufficienttoestimatethetermu−v¯forsomespecialelement¯v∈Vh.Forthiselementv¯∈Vh,wechoosetheinterpolantIh(u),whereIh:u∈C(Ω)¯u=0on∂Ω→Vh,(2.47)u→Ih(u)withIh(u)(ai)=u(ai).Thisinterpolantexistsandisunique(Lemma2.10).Obviously,minu−v1v∈Vh≤u−Ih(u)1foru∈C(Ω)and¯u=0on∂Ω.Iftheweaksolutionupossessesweakderivativesofsecondorder,thenforcertainsufficientlyfinetriangulationsTh,i.e.,0¯0,anestimateofthetypeu−Ih(u)1≤Ch(2.48)holds,whereCdependsonubutisindependentofh(cf.(3.88)).TheproofofthisestimatewillbeexplainedinSection3.4,wherealsosufficientconditionsonthefamilyoftriangulations(Th)hwillbespecified.Exercises12.4Leta(u,v):=x2u
v
dxforarbitraryu,v∈H1(0,1).00(a)ShowthatthereisnoconstantC1>0suchthattheinequality
1
21a(u,u)≥C1(u)dxforallu∈H0(0,1)0 2.3.StabilityandConvergence73isvalid.N(b)NowletTh:={(xi−1,xi)}i=1,N∈N,beanequidistantpartitionofN−1(0,1)withtheparameterh=1/NandVh:=span{ϕi}i=1,where(x−xi−1)/hin(xi−1,xi),ϕi(x):=(xi+1−x)/hin(xi,xi+1),0otherwise.DoesthereexistaconstantC2>0with
1
2a(uh,uh)≥C2(uh)dxforalluh∈Vh?02.5(a)ForΩ:=(α,β)×(γ,δ)andVaccordingto(2.7),provetheinequalityofPoincar´e:ThereexistsapositiveconstantCwithu0≤Cuaforallu∈V.xHint:Startwiththerelationu(x,y)=∂xu(s,y)ds.α(b)ForΩ:=(α,β)andv∈C([α,β])withapiecewisecontinuousderivativev
andv(γ)=0forsomeγ∈[α,β],showthatv≤(β−α)v
.002.6LetΩ:=(0,1)×(0,1).Givenf∈C(Ω),discretizetheboundaryvalueproblem−∆u=finΩ,u=0on∂Ω,bymeansoftheusualfive-pointdifferencestencilaswellasbymeansofthefiniteelementmethodwithlinearelements.AquadraticgridaswellasthecorrespondingFriedrichs–Kellertriangulationwillbeused.Provethefollowingstabilityestimatesforthematrixofthelinearsystemofequations:−11−11−1(a)Ah∞≤,(b)Ah2≤,(c)Ah0≤1,816where·∞,·2denotethemaximumrowsumnormandthespectralnormofamatrix,respectively,andA−1:=supv2/v2withh0vh∈Vhh0hav2:=|∇v|2dx.haΩhComment:Theconstantin(c)isnotoptimal.2.7LetΩbeadomainwithpolygonalboundaryandletThbeaconform-ingtriangulationofΩ.Thenodesaiofthetriangulationareenumeratedfrom1toM.Letthetriangulationsatisfythefollowingassumption:ThereexistconstantsC1,C2>0suchthatforalltrianglesK∈Ththerelation22C1h≤vol(K)≤C2h 742.FiniteElementMethodforPoissonEquationissatisfied.hdenotesthemaximumofthediametersofallelementsofTh.(a)Showtheequivalenceofthefollowingnormsforuh∈VhinthespaceVhofcontinuous,piecewiselinearfunctionsoverΩ:
1/2M1/222uh0:=|uh|dx,uh0,h:=huh(ai).Ωi=1(b)ConsiderthespecialcaseΩ:=(0,1)×(0,1)withtheFriedrichs–KellertriangulationaswellasthesubspaceV∩H1(Ω)andfind“asgoodh0aspossible”constantsinthecorrespondingequivalenceestimate.2.4TheImplementationoftheFiniteElementMethod:Part1Inthissectionwewillconsidersomeaspectsoftheimplementationofthefiniteelementmethodusinglinearansatzfunctionsontrianglesforthemodelboundaryvalueproblem(1.1),(1.2)onapolygonallyboundeddomainΩ⊂R2.ThecaseofinhomogeneousDirichletboundaryconditionswillbetreatedalsotoacertainextentasfarasitispossibleuptonow.2.4.1PreprocessorThemaintaskofthepreprocessoristodeterminethetriangulation.Aninputfilemighthavethefollowingformat:Letthenumberofvariables(includingalsotheboundarynodesforDirichletboundaryconditions)beM.Wegeneratethefollowinglist:x-coordinateofnode1y-coordinateofnode1......x-coordinateofnodeMy-coordinateofnodeMLetthenumberof(triangular)elementsbeN.Theseelementswillbelistedintheelement-nodetable.Here,everyelementischaracterizedbytheindicesofthenodescorrespondingtothiselementinawell-definedorder(e.g.,counterclockwise);cf.Figure2.11.117104Figure2.11.Elementno.10withnodesnos.4,11,7. 2.4.TheImplementationoftheFiniteElementMethod:Part175Forexample,the10throwoftheelement-nodetablecontainstheentry4117Usually,atriangulationisgeneratedbyatriangulationalgorithm.AshortoverviewonmethodsforthegridgenerationwillbegiveninSection4.1.Oneofthesimplestversionsofagridgenerationalgorithmhasthefollowingstructure(cf.Figure2.12):Figure2.12.Refinementbyquartering.Prescribeacoarsetriangulation(accordingtotheaboveformat)andrefinethistriangulation(repeatedly)bysubdividingatriangleinto4con-gruenttrianglesbyconnectingthemidpointsoftheedgeswithstraightlines.Ifthisuniformrefinementisdoneglobally,i.e.,foralltrianglesofthecoarsegrid,thentrianglesarecreatedthathavethesameinterioranglesastheelementsofthecoarsetriangulation.Thusthequalityofthetriangu-lation,indicated,forexample,bytheratiosofthediametersofanelementandofitsinscribedcircle(seeDefinition3.28),doesnotchange.However,ifthesubdivisionisperformedonlylocally,theresultingtriangulationisnolongeradmissible,ingeneral.Suchaninadmissibletriangulationcanbecorrectedbybisectionofthecorrespondingneighbouring(unrefined)tri-angles.Butthisimpliesthatsomeoftheinterioranglesarebisectedandconsequently,thequalityofthetriangulationbecomespoorerifthebisec-tionstepisperformedtoofrequently.Thefollowingalgorithmcircumventsthedepictedproblem.ItisduetoR.Bankandisimplemented,forexample,inthePLTMGcode(see[4]). 762.FiniteElementMethodforPoissonEquationAPossibleRefinementAlgorithmLeta(uniform)triangulationTbegiven(e.g.,byrepeateduniformrefine-mentofacoarsetriangulation).Theedgesofthistriangulationarecalledrededges.(1)Subdividetheedgesaccordingtoacertainlocalrefinementcriterion(introductionofnewnodes)bysuccessivebisection(cf.Figure2.13)...............Figure2.13.Newnodesonedges.(2)IfatriangleK∈Thasonitsedgesinadditiontotheverticestwoormorenodes,thensubdivideKintofourcongruenttriangles.Iterateoverstep2(cf.Figure2.14).(3)Subdividethetriangleswithnodesatthemidpointsoftheedgesinto2trianglesbybisection.Thisstepintroducestheso-calledgreenedges.(4)Iftherefinementistobecontinued,firstremovethegreenedges.2.4.2AssemblingDenotebyϕ1,...,ϕMtheglobalbasisfunctions.ThenthestiffnessmatrixAhhasthefollowingentries:
N(m)(Ah)ij=∇ϕj·∇ϕidx=AijΩm=1with
(m)Aij=∇ϕj·∇ϕidx.KmLeta1,...,aMdenotethenodesofthetriangulation.Becauseoftheimplication(m)Aij=0⇒ai,aj∈Km(m)(cf.(2.37)),theelementKmyieldsnonzerocontributionsforAijonlyifai,aj∈Kmatbest.SuchnonzerocontributionsarecalledelemententriesofAh.TheyadduptotheentriesofAh. 2.4.TheImplementationoftheFiniteElementMethod:Part177.................................:greenedgesFigure2.14.Tworefinementsequences.InExample2.12weexplainedanode-basedassemblingofthestiffnessmatrix.Incontrasttothisandonthebasisoftheaboveobservations,inthefollowingwewillperformanelement-basedassemblingofthestiffnessmatrix.ToassembletheentriesofA(m),wewillstartfromalocalnumbering(cf.Figure2.15)ofthenodesbyassigningthelocalnumbers1,2,3totheglobalnodenumbersr1,r2,r3(numberedcounterclockwise).Incontrasttotheusualnotationadoptedinthisbook,hereindicesofvectorsaccordingtothelocalnumberingareincludedinparenthesesandwrittenassuperscripts. 782.FiniteElementMethodforPoissonEquationr33Kmr11r22Figure2.15.Global(left)andlocalnumbering.Thusinfact,wegenerateA(m)asA˜(m).rirjiji,j=1,2,3i,j=1,2,3Todothis,wefirstperformatransformationofKmontosomereferenceelementandthenweevaluatetheintegralonthiselementexactly.Hencetheentryoftheelementstiffnessmatrixreadsas
A˜(m)=∇ϕ·∇ϕdx.ijrjriKmThereferenceelementKˆistransformedontotheglobalelementKmbymeansoftherelationF(ˆx)=Bxˆ+d,thereforeDxˆu(F(ˆx))=Dxu(F(ˆx))DxˆF(ˆx)=Dxu(F(ˆx))B,whereDxudenotestherowvector(∂1u,∂2u),i.e.,thecorrespondingdif-ferentialoperator.UsingthemorestandardnotationintermsofgradientsandtakingintoconsiderationtherelationB−T:=(B−1)T,weobtain∇u(F(ˆx))=B−T∇(u(F(ˆx)))(2.49)xxˆandthus
A˜(m)=∇ϕ(F(ˆx))·∇ϕ(F(ˆx))|det(DF(ˆx))|dxˆijxrjxriKˆ
=B−T∇ϕ(F(ˆx))·B−T∇(ϕ(F(ˆx)))|det(B)|dxˆxˆrjxˆriKˆ
=B−T∇ϕˆ(ˆx)·B−T∇ϕˆ(ˆx)|det(B)|dxˆ(2.50)xˆrjxˆriKˆ
=B−T∇N(ˆx)·B−T∇N(ˆx)|det(B)|dx,ˆxˆjxˆiKˆwherethetransformedbasisfunctionsˆϕri,ˆϕ(ˆx):=ϕ(F(ˆx))coincidewiththelocalbasisfunctionsonKˆ,i.e.,withtheshapefunctionsNi:ϕˆri(ˆx)=Ni(ˆx)forˆx∈K.ˆTheshapefunctionsNihavebeendefinedin(2.29)(where(x,y)theremustbereplacedby(ˆx1,xˆ2)here)forthestandardreferenceelementdefinedthere. 2.4.TheImplementationoftheFiniteElementMethod:Part179−1−1TT−1IntroducingthematrixC:=BB=BB,wecanwrite
A˜(m)=C∇N(ˆx)·∇N(ˆx)|det(B)|dx.ˆ(2.51)ijxˆjxˆiKˆDenotingthematrixBbyB=b(1),b(2),thenitfollowsthat−1b(1)·b(1)b(1)·b(2)1b(2)·b(2)−b(1)·b(2)C==b(1)·b(2)b(2)·b(2)det(B)2−b(1)·b(2)b(1)·b(1)becausedet(BTB)=det(B)2.Thepreviousconsiderationscanbeeas-ilyextendedtothecomputationofthestiffnessmatricesofmoregeneraldifferentialoperatorslike
K(x)∇ϕj(x)·∇ϕi(x)dxΩ(cf.Section3.5).Forthestandardreferenceelement,whichweusefromnowon,wehaveb(1)=a(2)−a(1),b(2)=a(3)−a(1).Herea(i),i=1,2,3,arethelocallynumberednodesofKinterpretedasvectorsofR2.Fromnowonwemakealsouseofthespecialformofthestiffnessmatrixandobtain
A˜(m)=γ∂N∂Ndxˆij1xˆ1jxˆ1iKˆ
+γ2∂xˆ1Nj∂xˆ2Ni+∂xˆ2Nj∂xˆ1Nidxˆ(2.52)Kˆ
+γ3∂xˆ2Nj∂xˆ2NidxˆKˆwith1(3)(1)(3)(1)γ1:=c11|det(B)|=a−a·a−a,|det(B)|1(2)(1)(3)(1)γ2:=c12|det(B)|=−a−a·a−a,|det(B)|1(2)(1)(2)(1)γ3:=c22|det(B)|=a−a·a−a.|det(B)|Intheimplementationitisadvisabletocomputethevaluesγijustoncefromthelocalgeometricalinformationgivenintheformoftheverticesa(i)=aandtostorethempermanently.riThusweobtainforthelocalstiffnessmatrixA˜(m)=γS+γS+γS(2.53)112233with
S1:=∂xˆ1Nj∂xˆ1Nidxˆ,Kˆij 802.FiniteElementMethodforPoissonEquation
S2:=∂xˆ1Nj∂xˆ2Ni+∂xˆ2Nj∂xˆ1Nidxˆ,Kˆij
S3:=∂xˆ2Nj∂xˆ2Nidxˆ.KˆijAnexplicitcomputationofthematricesSiispossiblebecausetheintegrandsareconstant,andalsothesematricescanbestoredpermanently:1−102−1−110−1111S1=−110,S2=−101,S3=000.222000−110−101Theright-handside(qh)i=Ωf(x)ϕi(x)dxcanbetreatedinasimilarmanner:N(q)=q(m)hiim=1with
q(m)=f(x)ϕ(x)dx(=0⇒a∈K).iiimKm(m)Again,wetransformtheglobalnumberingqriforthetrianglei=1,2,3(m)Km=conv{ar1,ar2,ar3}intothelocalnumberingq˜ii=1,2,3.Anal-ogouslytothedeterminationoftheentriesofthestiffnessmatrix,wehave
(m)q˜i=f(F(ˆx))ϕri(F(ˆx))|det(B)|dxˆKˆ
=fˆ(ˆx)Ni(ˆx)|det(B)|dx,ˆKˆwherefˆ(ˆx):=f(F(ˆx))forˆx∈K.ˆIngeneral,thisintegralcannotbeevaluatedexactly.Therefore,ithastobeapproximatedbyaquadraturerule.AquadratureruleforKˆg(ˆx)dxˆisofthetypeRωgˆb(k)kk=1withcertainweightsωandquadraturepointsˆb(k).Asanexample,wetakekthetrapezoidalrule(cf.(2.38)),whereˆb(1)=ˆa=(0,0),ˆb(2)=ˆa=(1,0),ˆb(3)=ˆa=(0,1),123ω=1,k=1,2,3.k6 2.4.TheImplementationoftheFiniteElementMethod:Part181Thusforarbitrarybutfixedquadraturerules,wehaveRq˜(m)≈ωfˆˆb(k)Nˆb(k)|det(B)|.(2.54)ikik=1Ofcourse,theapplicationofdifferentquadraturerulesondifferentelementsispossible,too.ThevaluesNiˆb(k),i=1,2,3,k=1,...,R,shouldbeevaluatedjustonceandshouldbestored.ThediscussionontheuseofquadratureruleswillbecontinuedinSections3.5.2and3.6.Insummary,thefollowingalgorithmprovidestheassemblingofthestiffnessmatrixandtheright-handside:Loopoverallelementsm=1,...,N:•Allocatingalocalnumberingtothenodesbasedontheelement-nodetable:1→r1,2→r2,3→r3.•AssemblingoftheelementstiffnessmatrixA˜(m)accordingto(2.51)or(2.53).Assemblingoftheright-handsideaccordingto(2.54).•Loopoveri,j=1,2,3:(m)(Ah)rirj:=(Ah)rirj+A˜ij,(m)(qh)ri:=(qh)ri+˜qi.Forthesakeofefficiencyofthisalgorithm,itisnecessarytoadjustthememorystructuretotheparticularsituation;wewillseehowthiscanbedoneinSection2.5.2.4.3RealizationofDirichletBoundaryConditions:Part1NodeswhereaDirichletboundaryconditionisprescribedmustbelabeledspecially,here,forinstance,bytheconventionM=M1+M2,wherethenodesnumberedfromM1+1toMcorrespondtotheDirichletboundarynodes.Inmoregeneralcases,otherrealizationsaretobepreferred.Inthefirststepofassemblingofstiffnessmatrixandtheloadvector,theDirichletnodesaretreatedlikealltheotherones.Afterthis,theDirichletnodesareconsideredseparately.Ifsuchanodehasthenumberj,theboundaryconditionisincludedbythefollowingprocedure:Replacethejthrowandthejthcolumn(forconservationofthesym-metry)ofAhbythejthunitvectorand(qh)jbyg(aj),ifu(x)=g(x)isprescribedforx∈∂Ω.Ifthejthcolumnisreplacedbytheunitvector,theright-handside(qh)ifori=jmustbemodifiedto(qh)i−(Ah)ijg(aj).Inotherwords,thecontributionscausedbytheDirichletboundaryconditionareincludedintotheright-handside.Thisisexactlytheeliminationthatledtotheform(1.10),(1.11)inChapter1. 822.FiniteElementMethodforPoissonEquation2.5SolvingSparseSystemsofLinearEquationsbyDirectMethodsLetAbeanM×Mmatrix.Givenavectorq∈RM,weconsiderthesystemoflinearequationsAξ=q.Thematricesarisingfromthefiniteelementdiscretizationaresparse;i.e.,theyhaveaboundednumberofnonzeroentriesperrowindependentofthedimensionofthesystemofequations.ForthesimpleexampleofSec-tion2.2,thisboundisdeterminedbythenumberofneighbouringnodes(see(2.37)).Methodsforsolvingsystemsofequationsshouldtakeadvan-tageofthesparsestructure.Foriterativemethods,whichwillbeexaminedinChapter5,thisiseasiertoreachthanfordirectmethods.Therefore,theimportanceofdirectmethodshasdecreased.Nevertheless,inadaptedformandforsmallormediumsizeproblems,theyarestillthemethodofchoice.EliminationwithoutPivotingusingBandStructureInthegeneralcase,wherethematrixAisassumedonlytobenonsingular,thereexistM×MmatricesP,L,UsuchthatPA=LU.HerePisapermutationmatrix,Lisascaledlowertriangularmatrix,andUisanuppertriangularmatrix;i.e.,theyhavetheform10u11uij....L=.,U=..lij10uMMThisdecompositioncorrespondstotheGaussianeliminationmethodwithpivoting.Themethodisveryeasyandhasfavourablepropertieswithre-specttothesparsestructure,ifpivotingisnotnecessary(i.e.,P=I,A=LU).ThenthematrixAiscalledLUfactorizable.DenotebyAktheleadingprincipalsubmatrixofAofdimensionk×k,i.e.,a11···a1k......Ak:=...,ak1···akkandsupposethatitalreadyhasbeenfactorizedasAk=LkUk.Thisisobviouslypossiblefork=1:A1=(a11)=(1)(a11).ThematrixAk+1canberepresentedintheformofablockmatrixAkbAk+1=cTd 2.5.DirectMethodsforSparseLinearSystems83withb,c∈Rk,d∈R.UsingtheansatzLk0UkuLk+1=,Uk+1=lT10swithunknownvectorsu,l∈Rkands∈R,itfollowsthatA=LU⇐⇒Lu=b,UTl=c,lTu+s=d.(2.55)k+1k+1k+1kkFromthis,wehavethefollowingresult:LetAbenonsingular.ThenloweranduppertriangularmatricesL,UexistwithA=LUifandonlyifAkisnonsingularforall(2.56)1≤k≤M.Forthiscase,LandUaredetermineduniquely.Furthermore,from(2.55)wehavethefollowingimportantconsequences:Ifthefirstlcomponentsofthevectorbareequaltozero,thenthisisvalidforthevectoru,too:00Ifb=,thenualsohasthestructureu=.βSimilarly,00c=impliesthestructurel=.γλForexample,ifthematrixAhasastructureasshowninFigure2.16,thenthezerosoutsideofthesurroundedentriesarepreservedaftertheLUfactorization.Beforeweintroduceappropriatedefinitionstogeneralizetheseresults,wewanttoconsiderthespecialcaseofsymmetricmatrices.|∗|0|∗|000|∗∗|0|∗|A=|∗∗∗∗∗|00|∗∗0|0|∗∗0∗|Figure2.16.Profileofamatrix.IfAisasbeforenonsingularandLUfactorizable,thenU=DLTwithadiagonalmatrixD=diag(di),andthereforeA=LDLT.ThisistruebecauseAhastheformA=LDU˜,wheretheuppertriangularmatrixU˜satisfiesthescalingcondition˜uii=1foralli=1,...,M.Suchafactorizationisunique,andthusTTTA=AimpliesL=U,˜thereforeA=LDL. 842.FiniteElementMethodforPoissonEquationIfinparticularAissymmetricandpositivedefinite,thenalsodi>0isvalid.ThusexactlyonematrixL˜oftheforml110L˜=..withl.ii>0forallilijlMMexistssuchthatA=L˜L˜T,theso-calledCholeskydecomposition.Wehave√√"L˜Chol=LGaussD,whereD:=diagdi.ThisshowsthattheCholeskymethodforthedeterminationoftheCholeskyfactorL˜alsopreservescertainzerosofAinthesamewayastheGaussianeliminationwithoutpivoting.Inwhatfollows,wewanttospecifythesetofzerosthatispreservedbyGaussianeliminationwithoutpivoting.Wewillnotconsiderasymmetricmatrix;butforthesakeofsimplicitywewillconsideramatrixwithasymmetricdistributionofitsentries.Definition2.19LetA∈RM×Mbeamatrixsuchthata=0fori=ii1,...,Mandaij=0ifandonlyifaji=0foralli,j=1,...,M.(2.57)Wedefine,fori=1,...,M,fi(A):=minjaij=0,1≤j≤i.Thenmi(A):=i−fi(A)iscalledtheith(left-handside)rowbandwidthofA.ThebandwidthofamatrixAthatsatisfies(2.57)isthenumberm(A):=maxmi(A)=maxi−jaij=0,1≤j≤i≤M.1≤i≤MThebandofthematrixAisB(A):=(i,j),(j,i)i−m(A)≤j≤i,1≤i≤M.ThesetEnv(A):=(i,j),(j,i)fi(A)≤j≤i,1≤i≤MiscalledthehullorenvelopeofA.ThenumberMp(A):=M+2mi(A)i=1iscalledtheprofileofA. 2.5.DirectMethodsforSparseLinearSystems85TheprofileisthenumberofelementsofEnv(A).ForthematrixAinFigure2.16wehave(m1(A),...,m5(A))=(0,0,2,1,3),m(A)=3,andp(A)=17.Summarizingtheaboveconsiderations,wehaveprovedthefollowingtheorem:Theorem2.20LetAbeamatrixwiththesymmetricstructure(2.57).ThentheCholeskymethodortheGaussianeliminationwithoutpivotingpreservesthehullandinparticularthebandwidth.Thehullmaycontainzerosthatwillbereplacedby(nonzero)entriesduringthedecompositionprocess.Therefore,inordertokeepthisfill-insmall,theprofileshouldbeassmallaspossible.Furthermore,inordertoexploitthematrixstructureforanefficientassemblingandstorage,thisstructure(orsomeestimateofit)shouldbeknowninadvance,beforethecomputationofthematrixentriesisstarted.Forexample,ifAisastiffnessmatrixwiththeentries
aij=a(ϕj,ϕi)=∇ϕj·∇ϕidx,Ωthenthepropertyaij=0⇒ai,ajareneighbouringnodescanbeusedforthedefinitionofan(eventuallytoolarge)symmetricmatrixstructure.Thisisalsovalidforthecaseofanonsymmetricbilinearformandthusanonsymmetricstiffnessmatrix.Alsointhiscase,thedefinitionoffi(A)canbereplacedbyfi(A):=minj1≤j≤i,jisaneighbouringnodeofi.Sincethecharacterization(2.56)ofthepossibilityoftheGaussianelim-inationwithoutpivotingcannotbecheckeddirectly,wehavetospecifysufficientconditions.Examplesforsuchconditionsarethefollowing(see[34]):•Aissymmetricandpositivedefinite,•AisanM-matrix.∗Sufficientconditionsforthispropertyweregivenin(1.32)and(1.32).InSection3.9,geometricalconditionsforthefamilyoftriangula-tions(Th)hwillbederivedthatguaranteethatthefiniteelementdiscretizationconsideredherecreatesanM-matrix.DataStructuresForsparsematrices,itisappropriatetostoreonlythecomponentswithinthebandorthehull.AsymmetricmatrixA∈RM×MwithbandwidthmcanbestoredinM(m+1)memorypositions.Bymeansoftheindex 862.FiniteElementMethodforPoissonEquationconversionaikbi,k−i+m+1fork≤i,thematrixa11a12···a1,m+1....a21a22···..0........................M×MA=am+1,1am+1,2···am+1,m+1...∈R............................0.....aM,M−m···aM,M−1aM,Mismappedtothematrix0······0a110···0a21a22.........0am,1······am,mB=∈RM×(m+1).am+1,1······am+1,mam+1,m+1..............................aM,M−m······aM,M−1aM,MTheunusedelementsofB,i.e.,(B)ijfori=1,...,m,j=1,...,m+1−i,areherefilledwithzeros.Forageneralbandmatrix,thematrixB∈RM×(2m+1)obtainedbytheaboveconversionhasthefollowingform:0···0a11a12···a1,m+10···a21a22······a2,m+2..................0am,1············am,2mam+1,1···············am+1,2m+1B=......................aM−m,M−2m···············aM−m,MaM−m+1,M−2m+1············aM−m+1,M0..................aM,M−m······aM,M0···0Here,intherightlowerpartofthematrix,afurthersectorofunusedelementsarose,whichisalsofilledwithzeros.Ifthestorageisbasedonthehull,additionallyapointerfieldisneeded,whichpointstothediagonalelements,forexample.Ifthematrixissym- 2.5.DirectMethodsforSparseLinearSystems87metric,againthestorageofthelowertriangularmatrixissufficient.ForthematrixAfromFigure2.16undertheassumptionthatAissymmetric,thepointerfieldcouldactasshowninFigure2.17.a11a22a31a32a33a43a44a52a53a54a55 125711Figure2.17.Linearstorageofthehull.CoupledAssemblingandDecompositionAformerlypopularmethod,theso-calledfrontalmethod,performssimultaneouslyassemblingandtheCholeskyfactorization.WeconsiderthismethodfortheexampleofthestiffnessmatrixAh=(a)∈RM×Mwithbandwidthm(withtheoriginalnumbering).ijThemethodisbasedonthekthstepoftheGaussianorCholeskymethod(cf.Figure2.18).k0kBk0Figure2.18.kthstepoftheCholeskymethod.OnlytheentriesofBkaretobechanged,i.e.,onlythoseelementsaijwithk≤i,j≤k+m.Thecorrespondingformulais(k)(k+1)(k)aik(k)a=a−a,i,j=k+1,...,k+m.(2.58)ijij(k)kjakkHere,theupperindicesindicatethestepsoftheeliminationmethod,whichwestoreinaij.Theentriesaijaregeneratedbysummationofentriesof 882.FiniteElementMethodforPoissonEquationtheelementstiffnessmatrixofthoseelementsKthatcontainnodeswiththeindicesi,j.(k)(k)Furthermore,toperformtheeliminationstep(2.58),onlya,aforikkj(k)i,j=k,...,k+mmustbecompletelyassembled;aij,i,j=k+1,...,k+(k)(k+1)(k+1)(k+1)m,canbereplacedby˜aijifaijislaterdefinedbyaij:=˜aij+(k)(k)aij−a˜ij.Thatis,forthepresent,aijneedstoconsistofonlyafewcontributionsofelementsKwithnodesi,jinK.Fromtheseobservations,thefollowingalgorithmisobtained.Thekthstepfork=1,...,Mreadsasfollows:•AssembleallofthemissingcontributionsofelementsKthatcontainthenodewithindexk.•ComputeA(k+1)bymodificationoftheentriesofBaccordingtok(2.58).•StorethekthrowofA(k+1),alsooutofthemainmemory.•DefineBk+1(byasouth-eastshift).Heretheassemblingisnode-basedandnotelement-based.TheadvantageofthismethodisthatAhneednotbecompletelyassem-bledandstoredinthemainmemory,butonlyamatrixB∈R(m+1)×(m+1).kOfcourse,ifMisnottoolarge,theremaybenoadvantage.BandwidthReductionThecomplexity,i.e.,thenumberofoperations,iscrucialfortheapplicationofaparticularmethod:TheCholeskymethod,appliedtoasymmetricmatrixA∈RM×Mwithbandwidthm,requiresO(m2M)operationsinordertocomputeL.However,thebandwidthmofthestiffnessmatrixdependsonthenum-beringofthenodes.Therefore,anumberingistobefoundwherethenumbermisassmallaspossible.WewanttoconsiderthisagainfortheexampleofthePoissonequationontherectanglewiththediscretizationaccordingtoFigure2.9.Lettheinte-riornodeshavethecoordinates(ih,jh)withi=1,...k−1,j=1,...,l−1.Thediscretizationcorrespondstothefinitedifferencemethodintroducedbeginningwith(1.10);i.e.,thebandwidthisequaltok−1forarowwisenumberingorl−1foracolumnwisenumbering.Forklorkl,thisfactresultsinalargedifferenceofthebandwidthmoroftheprofile(ofthelefttriangle),whichisofsize(k−1)(l−1)(m+1)exceptforatermofm2.Therefore,thecolumnwisenumberingispreferredforkl;therowwisenumberingispreferredforkl.ForageneraldomainΩ,anumberingalgorithmbasedonagiventri-angulationThandonabasis{ϕi}ofVhisnecessarywiththefollowingproperties: 2.5.DirectMethodsforSparseLinearSystems89ThestructureofAresultingfromthenumberingmustbesuchthatthebandortheprofileofAisassmallaspossible.Furthermore,thenumberingalgorithmshouldyieldthenumbersm(A)orfi(A),mi(A)suchthatthematrixAcanalsobeassembledusingtheelementmatricesA(k).GivenatriangulationThandacorrespondingbasisϕi1≤i≤MofVh,westartwiththeassignmentofsomegraphGtothistriangulationasfollows:ThenodesofGcoincidewiththenodes{a1,...,aM}ofthetriangulation.Thedefinitionofitsedgesis:(ai,aj)isanedgeofG⇐⇒thereexistsaK∈Thsuchthatϕi|K≡0,ϕj|K≡0.InFigure2.19someexamplesaregiven,wheretheexample(2)willbeintroducedinSection3.3.triangulation:......(1)(2).....linearansatzontriangle(bi)linearansatzonquadrilateralGraph:...........Figure2.19.Triangulationandassignedgraph.Ifseveraldegreesoffreedomareassignedtosomenodeofthetriangu-lationTh,thenalsoinGseveralnodesareassignedtoit.Thisisthecase,forexample,ifso-calledHermiteelementsareconsidered,whichwillbeintroducedinSection3.3.Thecostsofadministrationaresmallifthesamenumberofdegreesoffreedomisassignedtoallnodesofthetriangulation.Anoften-usednumberingalgorithmistheCuthill–McKeemethod.ThisalgorithmoperatesonthegraphGjustdefined.Twonodesai,ajofGarecalledneighbouredif(ai,aj)isanedgeofG.ThedegreeofanodeaiofGisdefinedasthenumberofneighboursofai.Thekthstepofthealgorithmfork=1,...,Mhasthefollowingform:k=1:Chooseastartingnode,whichgetsthenumber1.Thisstartingnodeformsthelevel1.k>1:Ifallnodesarealreadynumbered,thealgorithmisterminated.Otherwise,thelevelkisformedbytakingallthenodesthatarenotnum- 902.FiniteElementMethodforPoissonEquationberedyetandthatareneighboursofanodeoflevelk−1.Thenodesoflevelkwillbeconsecutivelynumbered.Withinalevel,wecansort,forexample,bythedegree,wherethenodewiththesmallestdegreeisnumberedfirst.ThereverseCuthill–McKeemethodconsistsoftheabovemethodandtheinversionofthenumberingattheend;i.e.,newnodenumber=M+1−oldnodenumber.Thiscorrespondstoareflectionofthematrixatthecounterdiagonal.Thebandwidthdoesnotchangebytheinversion,buttheprofilemaydiminishdrasticallyformanyexamples(cf.Figure2.20).************************************Figure2.20.Changeofthehullbyreflectionatthecounterdiagonal.ThefollowingestimateholdsforthebandwidthmofthenumberingcreatedbytheCuthill–McKeealgorithm:D+i≤m≤max(Nk−1+Nk−1).22≤k≤νHereDisthemaximumdegreeofanodeofG,νisthenumberoflevels,andNkisthenumberofnodesoflevelk.Thenumberiisequalto0ifDiseven,andiisequalto1ifDisodd.Theleft-handsideoftheaboveinequalityiseasytounderstandbymeansofthefollowingargument:Toreachaminimalbandwidth,allnodesthatareneighboursofaiinthegraphGshouldalsobeneighboursofaiinthenumbering.Thenthebestsituationisgiveniftheneighbourednodeswouldappearuniformlyimmediatelybeforeandafterai.IfDisodd,thenonesidehasonenodemorethantheother.Toverifytheright-handside,consideranodeaithatbelongstolevelk−1aswellasanodeajthatisaneighbourofaiinthegraphGandthatisnotyetnumberedinlevelk−1.Therefore,ajwillgetanumberinthekthstep.Thelargestbandwidthisobtainedifaiisthefirstnodeofthenumberingoflevelk−1andifajisthelastnodeoflevelk.Henceexactly(Nk−1−1)+(Nk−1)nodesliebetweenbothofthese;i.e.,theirdistanceinthenumberingisNk−1+Nk−1.ItisfavourableifthenumberνoflevelsisaslargeaspossibleandifallthenumbersNkareofthesamesize,ifpossible.Therefore,thestartingnodeshouldbechosen“atoneend”ofthegraphGifpossible;ifallthe 2.5.DirectMethodsforSparseLinearSystems91startingnodesaretobechecked,theexpensewillbeO(MM˜),whereM˜isthenumberofedgesofG.Onepossibilityconsistsinchoosinganodewithminimumdegreeforthestartingnode.Anotherpossibilityistoletthealgorithmrunonceandthentochoosethelast-numberednodeasthestartingnode.Ifanumberingiscreatedbythe(reverse)Cuthill–McKeealgorithm,wecantrytoimproveit“locally”,i.e.,byexchangingparticularnodes.Exercise2.8ShowthatthenumberofarithmeticoperationsfortheCholeskymethodforanM×MmatrixwithbandwidthmhasorderMm2/2;additionally,Msquarerootshavetobecalculated. 3TheFiniteElementMethodforLinearEllipticBoundaryValueProblemsofSecondOrder3.1VariationalEquationsandSobolevSpacesWenowcontinuethedefinitionandanalysisofthe“correct”functionspacesthatwebeganin(2.17)–(2.20).Anessentialassumptionensuringtheexis-tenceofasolutionofthevariationalequation(2.13)isthecompletenessofthebasicspace(V,·).IntheconcretecaseofthePoissonequationthe“preliminary”functionspaceVaccordingto(2.7)canbeequippedwiththenorm·1,definedin(2.19),whichhasbeenshowntobeequivalenttothenorm·a,givenin(2.6)(see(2.46)).Ifweconsidertheminimizationproblem(2.14),whichisequivalenttothevariationalequation,thefunc-tionalFisboundedfrombelowsuchthattheinfimumassumesafinitevalueandthereexistsaminimalsequence(vn)ninV,thatis,asequencewiththepropertylimF(vn)=infF(v)v∈V.n→∞TheformofFalsoimpliesthat(vn)nisaCauchysequence.Ifthissequenceconvergestoanelementv∈V,then,duetothecontinuityofFwithrespectto·,itfollowsthatvisasolutionoftheminimizationproblem.ThiscompletenessofVwithrespectto·a,andhencewithrespectto·1,isnotsatisfiedinthedefinition(2.7),asExample2.8hasshown.Therefore,anextensionofthebasicspaceV,asformulatedin(2.20),isnecessary.Thisspacewillturnouttobe“correct,”sinceitiscompletewithrespectto·1. 3.1.VariationalEquationsandSobolevSpaces93Inwhatfollowsweusethefollowinggeneralassumption:Visavectorspacewithscalarproduct·,·andthenorm·1/2inducedby·,·(forthis,v:=v,vforv∈Vissatisfied);Viscompletewithrespectto·,i.e.aHilbertspace;(3.1)a:V×V→Risa(notnecessarilysymmetric)bilinearform;b:V→Risalinearform.Thefollowingtheoremgeneralizestheaboveconsiderationtononsym-metricbilinearforms:Theorem3.1(Lax–Milgram)Supposethefollowingconditionsaresat-isfied:•aiscontinuous(cf.(2.42));thatis,thereexistssomeconstantM>0suchthat|a(u,v)|≤Muvforallu,v∈V;(3.2)•aisV-elliptic(cf.(2.43));thatis,thereexistssomeconstantα>0suchthata(u,u)≥αu2forallu∈V;(3.3)•biscontinuous;thatis,thereexistssomeconstantC>0suchthat|b(u)|≤Cuforallu∈V.(3.4)Thenthevariationalequation(2.21),namely,findu¯∈Vsuchthata(¯u,v)=b(v)forallv∈V,(3.5)hasoneandonlyonesolution.Here,onecannotavoidtheassumptions(3.1)and(3.2)–(3.4)ingeneral.Proof:See,forexample,[26];foranalternativeproofseeExercise3.1.
Nowreturningtotheexampleabove,theassumptions(3.2)and(3.3)areobviouslysatisfiedfor·=·a.However,the“preliminary”definitionofthefunctionspaceVof(2.7)withnorm·adefinedin(2.19)isinsufficient,since(V,·a)isnotcomplete.Therefore,thespaceVmustbeextended.Indeed,itisnotthenormonVthathasbeenchosenincorrectly,sinceVisalsonotcompletewithrespecttoanothernorm·thatsatisfies(3.2)and(3.3).Inthiscasethenorms·and·awouldbeequivalent(cf.(2.46)),andconsequently,(V,·a)complete⇐⇒(V,·)complete.NowweextendthespaceVandtherebygeneralizedefinition(2.17). 943.FiniteElementMethodsforLinearEllipticProblemsDefinition3.2SupposeΩ⊂Rdisa(bounded)domain.TheSobolevspaceHk(Ω)isdefinedbyHk(Ω):=v:Ω→Rv∈L2(Ω),theweakderivatives∂αvexistinL2(Ω)andforallmulti-indicesαwith|α|≤k.Ascalarproduct·,·andtheresultingnorm·inHk(Ω)aredefinedkkasfollows:
v,w:=∂αv∂αwdx,(3.6)kΩαmulti−index|α|≤k
1/21/2∂α2vk:=v,vk=vdx(3.7)Ωαmulti−index|α|≤k
1/21/2α2α2=∂vdx=∂v.0Ωαmulti−indexαmulti−index|α|≤k|α|≤kGreaterflexibilitywithrespecttothesmoothnesspropertiesofthefunc-tionsthatarecontainedinthedefinitionisobtainedbyrequiringthatvanditsweakderivativesshouldbelongnottoL2(Ω)buttoLp(Ω).Inthenormdenotedby·theL2(Ω)andnorms(forthevectorofthek,p2derivativenorms)havetobereplacedbytheLp(Ω)andnorms,respec-ptively(seeAppendicesA.3andA.5).However,theresultingspace,denotedbyWk(Ω),cannolongerbeequippedwithascalarproductforp=2.Al-pthoughthesespacesoffergreaterflexibility,wewillnotusethemexceptinSections3.6,6.2,and9.3.Besidesthenorms·,thereareseminorms|·|for0≤l≤kinHk(Ω),kldefinedby1/22|v|=∂αv,l0αmulti−index|α|=lsuchthat1/2kv=|v|2,kll=0Inparticular,thesedefinitionsarecompatiblewiththosein(2.18),
v,w:=vw+∇v·∇wdx,1Ωandwiththenotation·fortheL2(Ω)norm,givingameaningtothis0one. 3.1.VariationalEquationsandSobolevSpaces95Theabovedefinitioncontainssomeassertionsthatareformulatedinthefollowingtheorem:Theorem3.3Thebilinearform·,·isascalarproductonHk(Ω);thatkis,·isanormonHk(Ω).kHk(Ω)iscompletewithrespectto·,andisthusaHilbertspace.kProof:See,forexample,[37].
Obviously,Hk(Ω)⊂Hl(Ω)fork≥l,andtheembeddingiscontinuous,sincekvl≤vkforallv∈H(Ω).(3.8)Intheone-dimensionalcase(d=1)v∈H1(Ω)isnecessarilycontinuous:Lemma3.4H1(a,b)⊂C[a,b],andtheembeddingiscontinuous,whereC[a,b]isequippedwiththenorm·∞;thatis,thereexistssomeconstantC>0suchthat1v∞≤Cv1forallv∈H(a,b).(3.9)Proof:SeeExercise3.2.
SincetheelementsofHk(Ω)arefirstofallonlysquareintegrablefunc-tions,theyaredeterminedonlyuptopointsofasetof(d-dimensional)measurezero.Therefore,aresultasinLemma3.4meansthatthefunc-tionisallowedtohaveremovablediscontinuitiesatpointsofsuchasetofmeasurezerothatvanishbymodifyingthefunctionvalues.However,ingeneral,H1(Ω)⊂C(Ω).¯Asanexampleforthis,weconsideracirculardomainindimensiond=2:Ω=B(0)=x∈R2|x|0suchthatγ(v)≤Cvforallv∈H1(Ω).001Hereγ(v)∈L2(∂Ω)iscalledthetraceofv∈H1(Ω).0Themappingγ0isnotsurjective;thatis,γ0(v)v∈H1(Ω)isarealsubsetofL2(∂Ω).Forallv∈C∞(Rd)|wehaveΩγ0(v)=v|∂Ω.Inthefollowingwewilluseagainv|∂Ωor“von∂Ω”forγ0(v),butinthesenseofTheorem3.5.Accordingtothistheorem,definition(2.20)iswell-definedwiththeinterpretationofuon∂Ωasthetrace:Definition3.6H1(Ω):=v∈H1(Ω)γ(v)=0(asafunctionon∂Ω).00Theorem3.7SupposeΩ⊂RdisaboundedLipschitzdomain.ThenC∞(Ω)isdenseinH1(Ω).00Proof:See[37].
TheassertionofTheorem3.5,thatC∞(Rd)|isdenseinH1(Ω),hasΩsevereconsequencesforthetreatmentoffunctionsinH1(Ω)whichareingeneralnotverysmooth.Itispossibletoconsiderthemassmoothfunctionsifattheendonlyrelationsinvolvingcontinuousexpressionsin·1(andnotrequiringsomethinglike∂iv∞)arise.Then,bysome“densityargument”theresultcanbetransferredtoH1(Ω)or,asforthetraceterm,newtermscanbedefinedforfunctionsinH1(Ω).Thus,fortheproofofLemma3.4itisnecessarysimplytoverifyestimate(3.9),forexampleforv∈C1[a,b].ByvirtueofTheorem3.7,analogousresultsholdforH1(Ω).0Hence,forv∈H1(Ω)integrationbypartsispossible:Theorem3.8SupposeΩ⊂RdisaboundedLipschitzdomain.Theouterunitnormalvectorν=(ν):∂Ω→Rdisdefinedalmosteverywhereii=1,...,dandν∈L∞(∂Ω).i 983.FiniteElementMethodsforLinearEllipticProblemsForv,w∈H1(Ω)andi=1,...,d,


∂ivwdx=−v∂iwdx+vwνidσ.ΩΩ∂ΩProof:See,forexample,[14]or[37].
Ifv∈H2(Ω),thenduetotheabovetheorem,v|:=γ(v)∈L2(∂Ω)∂Ω0and∂v|:=γ(∂v)∈L2(∂Ω),sincealso∂v∈H1(Ω).Hence,thenormali∂Ω0iiderivatived∂νv|∂Ω:=∂iv|∂Ωνii=1iswell-definedandbelongstoL2(∂Ω).Thus,thetracemappingγ:H2(Ω)→L2(∂Ω)×L2(∂Ω),v→(v|∂Ω,∂νv|∂Ω),iswell-definedandcontinuous.Thecontinuityofthismappingfollowsfromthefactthatitisacompositionofcontinuousmappings:2continuous1continuous2v∈H(Ω)→∂iv∈H(Ω)→∂iv|∂Ω∈L(∂Ω)continuous2→∂iv|∂Ωνi∈L(∂Ω).Corollary3.9SupposeΩ⊂RdisaboundedLipschitzdomain.(1)Letw∈H1(Ω),q∈H1(Ω),i=1,...,d.Theni


q·∇wdx=−∇·qwdx+q·νwdσ.(3.10)ΩΩ∂Ω(2)Letv∈H2(Ω),w∈H1(Ω).Then


∇v·∇wdx=−∆vwdx+∂νvwdσ.ΩΩ∂ΩTheintegrationbypartsformulasalsoholdmoregenerallyifonlyitisensuredthatthefunctionwhosetracehastobeformedbelongstoH1(Ω).Forexample,ifK=(k),wherek∈W1(Ω)andv∈H2(Ω),w∈ijijij∞H1(Ω),itfollowsthat


K∇v·∇wdx=−∇·(K∇v)wdx+K∇v·νwdσ(3.11)ΩΩ∂Ωwithconormalderivative(see(0.41))dT∂νKv:=K∇v·ν=∇v·Kν=kij∂jvνi.i,j=1 Exercises99HereitisimportantthatthecomponentsofK∇vbelongtoH1(Ω),usingthefactthatforv∈L2(Ω),k∈L∞(Ω),kv∈L2(Ω)andkv≤kv.0∞0Theorem3.10SupposeΩ⊂RdisaboundedLipschitzdomain.Ifk>d/2,thenHk(Ω)⊂C(Ω)¯,andtheembeddingiscontinuous.Proof:See,forexample,[37].
Fordimensiond=2thisrequiresk>1,andfordimensiond=3weneedk>3.Therefore,inbothcasesk=2satisfiestheassumptionofthe2abovetheorem.Exercises3.1ProvetheLax–MilgramTheoreminthefollowingway:(a)Show,byusingtheRieszrepresentationtheorem,theequivalenceof(3.5)withtheoperatorequationAu¯=fforA∈L[V,V]andf∈V.(b)Show,forTε∈L[V,V],Tεv:=v−ε(Av−f)andε>0,thatforsomeε>0,theoperatorTεisacontractiononV.ThenconcludetheassertionbyBanach’sfixed-pointtheorem(intheBanachspacesetting,cf.Remark8.5).3.2Proveestimate(3.9)byshowingthatevenforv∈H1(a,b),|v(x)−v(y)|≤|v||x−y|1/2forx,y∈(a,b).13.3SupposeΩ⊂R2istheopendiskwithradius1andcentre0.Prove2thatforthefunctionu(x):=ln|x|α,x∈Ω{0},α∈(0,1)wehave2u∈H1(Ω),butucannotbeextendedcontinuouslytox=0.3.4SupposeΩ⊂R2istheopenunitdisk.Provethateachu∈H1(Ω)√4hasatraceu|∂Ω∈L2(∂Ω)satisfyingu0,∂Ω≤8u1,Ω. 1003.FiniteElementMethodsforLinearEllipticProblems3.2EllipticBoundaryValueProblemsofSecondOrderInthissectionweintegrateboundaryvalueproblemsforthelinear,sta-tionarycaseofthedifferentialequation(0.33)intothegeneraltheoryofSection3.1.ConcerningthedomainwewillassumethatΩisaboundedLipschitzdomain.Weconsidertheequation(Lu)(x):=−∇·(K(x)∇u(x))+c(x)·∇u(x)+r(x)u(x)=f(x)forx∈Ω(3.12)withthedataK:Ω→Rd,d,c:Ω→Rd,r,f:Ω→R.AssumptionsabouttheCoefficientsandtheRight-HandSideForaninterpretationof(3.12)intheclassicalsense,weneed∂ikij,ci,r,f∈C(Ω)¯,i,j∈{1,...,d},(3.13)andforaninterpretationinthesenseofL2(Ω)withweakderivatives,andhenceforasolutioninH2(Ω),∂k,c,r∈L∞(Ω),f∈L2(Ω),i,j∈{1,...,d}.(3.14)iijiOncewehaveobtainedthevariationalformulation,weakerassumptionsaboutthesmoothnessofthecoefficientswillbesufficientfortheverifica-tionoftheproperties(3.2)–(3.4),whicharerequiredbytheLax–Milgram,namely,k,c,∇·c,r∈L∞(Ω),f∈L2(Ω),i,j∈{1,...,d},iji(3.15)andif|Γ∪Γ|>0,ν·c∈L∞(Γ∪Γ).12d−112Herewerefertoadefinitionoftheboundaryconditionsasin(0.36)–(0.39)(seealsobelow).Furthermore,theuniformellipticityofLisassumed:Thereexistssomeconstantk0>0suchthatfor(almost)everyx∈Ω,dk(x)ξξ≥k|ξ|2forallξ∈Rd(3.16)ijij0i,j=1(thatis,thecoefficientmatrixKispositivedefiniteuniformlyinx).Moreover,Kshouldbesymmetric.IfKisadiagonalmatrix,thatis,kij(x)=ki(x)δij(thisisinparticularthecaseifki(x)=k(x)withk:Ω→R,i∈{1,...,d},whereK∇ubecomesk∇u),thismeansthat(3.16)⇔ki(x)≥k0for(almost)everyx∈Ω,i∈{1,...,d}. 3.2.EllipticBoundaryValueProblems101Finally,thereexistsaconstantr0≥0suchthat1r(x)−∇·c(x)≥r0for(almost)everyx∈Ω.(3.17)2BoundaryConditionsAsinSection0.5,supposeΓ1,Γ2,Γ3isadisjointdecompositionoftheboundary∂Ω(cf.(0.39)):∂Ω=Γ1∪Γ2∪Γ3,whereΓ3isaclosedsubsetoftheboundary.Forgivenfunctionsgj:Γj→R,j=1,2,3,andα:Γ2→Rweassumeon∂Ω•Neumannboundarycondition(cf.(0.41)or(0.36))K∇u·ν=∂νKu=g1onΓ1,(3.18)•mixedboundarycondition(cf.(0.37))K∇u·ν+αu=∂νKu+αu=g2onΓ2,(3.19)•Dirichletboundarycondition(cf.(0.38))u=g3onΓ3.(3.20)Concerningtheboundarydatathefollowingisassumed:Fortheclassicalapproachweneedgj∈C(Γj),j=1,2,3,α∈C(Γ2),(3.21)whereasforthevariationalinterpretation,2∞gj∈L(Γj),j=1,2,3,α∈L(Γ2)(3.22)issufficient.3.2.1VariationalFormulationofSpecialCasesThebasicstrategyforthederivationofthevariationalformulationofboundaryvalueproblems(3.12)hasalreadybeendemonstratedinSec-tion2.1.Assumingtheexistenceofaclassicalsolutionof(3.12)thefollowingstepsareperformedingeneral:Step1:MultiplicationofthedifferentialequationbytestfunctionsthatarechosencompatiblewiththetypeofboundaryconditionandsubsequentintegrationoverthedomainΩ.Step2:Integrationbypartsunderincorporationoftheboundarycondi-tionsinordertoderiveasuitablebilinearform.Step3:Verificationoftherequiredpropertieslikeellipticityandcontinuity. 1023.FiniteElementMethodsforLinearEllipticProblemsInthefollowingtheabovestepswillbedescribedforsomeimportantspecialcases.(I)HomogeneousDirichletBoundaryCondition∂Ω=Γ,g≡0,V:=H1(Ω)330Supposeuisasolutionof(3.12),(3.20);thatis,inthesenseofclassicalsolutionsletu∈C2(Ω)∩C(Ω)andthedifferentialequation(3.12)be¯satisfiedpointwiseinΩundertheassumptions(3.13)aswellasu=0pointwiseon∂Ω.However,theweakercaseinwhichu∈H2(Ω)∩VandthedifferentialequationissatisfiedinthesenseofL2(Ω),nowundertheassumptions(3.14),canalsobeconsidered.Multiplying(3.12)byv∈C∞(Ω)(intheclassicalcase)orbyv∈V,0respectively,thenintegratingbypartsaccordingto(3.11)andtakingintoaccountthatv=0on∂ΩbyvirtueofthedefinitionofC∞(Ω)andH1(Ω),00respectively,weobtain
a(u,v):={K∇u·∇v+c·∇uv+ruv}dx(3.23)Ω
=b(v):=fvdxforallv∈C∞(Ω)orv∈V.0ΩThebilinearformaissymmetricifcvanishes(almosteverywhere).Forf∈L2(Ω),biscontinuouson(V,·1).(3.24)ThisfollowsdirectlyfromtheCauchy–Schwarzinequality,since
|b(v)|≤|f||v|dx≤f0v0≤f0v1forv∈V.ΩFurther,by(3.15),aiscontinuous(V,·1).(3.25)Proof:First,weobtain
|a(u,v)|≤{|K∇u||∇v|+|c||∇u||v|+|r||u||v|}dx.ΩHere|·|denotestheabsolutevalueofarealnumberortheEuclideannormofavector.Usingalso·2forthe(associated)spectralnorm,and·fortheL∞(Ω)normofafunction,wefurtherintroducethefollowing∞notations:*+C1:=maxK2∞,r∞<∞,C2:=|c|∞<∞.Byvirtueof|K(x)∇u(x)|≤K(x)2|∇u(x)|, 3.2.EllipticBoundaryValueProblems103wecontinuetoestimateasfollows:

|a(u,v)|≤C1{|∇u||∇v|+|u||v|}dx+C2|∇u||v|dx.ΩΩ!!=:A1=:A2TheintegrandofthefirstaddendisestimatedbytheCauchy–SchwarzinequalityforR2,andthentheCauchy–SchwarzinequalityforL2(Ω)isapplied:
221/2221/2A1≤C1|∇u|+|u||∇v|+|v|dxΩ
1/2
1/2≤C|u|2+|∇u|2dx|v|2+|∇v|2dx=Cuv.1111ΩΩDealingwithA,wecanemploytheCauchy–SchwarzinequalityforL2(Ω)2directly:
1/2
1/2A≤C|∇u|2dx|v|2dx22ΩΩ≤C2u1v0≤C2u1v1forallu,v∈V.Thus,theassertionfollows.
Remark3.11Intheproofofthepropositions(3.24)and(3.25)ithasnotbeenusedthatthefunctionsu,vsatisfyhomogeneousDirichletboundaryconditions.Therefore,undertheassumptions(3.15)thesepropertiesholdforeverysubspaceV⊂H1(Ω).ConditionsfortheV-Ellipticityofa(A)aissymmetric;thatisc=0(a.e.):Condition(3.17)thenhasthesimpleformr(x)≥r0foralmostallx∈Ω.(A1)c=0,r0>0:Becauseof(3.16)wedirectlyget
a(u,u)≥{k|∇u|2+r|u|2}dx≥Cu2forallu∈V,0031ΩwhereC:=min{k,r}.ThisalsoholdsforeverysubspaceV⊂H1(Ω).300(A2)c=0,r0≥0:AccordingtothePoincar´einequality(Theorem2.18),thereexistssomeconstantC>0,independentofu,suchthatforu∈H1(Ω)P0
1/2u≤C|∇u|2dx.0PΩ 1043.FiniteElementMethodsforLinearEllipticProblemsTakingintoaccount(3.16)andusingthesimpledecompositionk0=kC20P2+2k0wecanfurtherconcludethat1+C1+CPP
a(u,u)≥k|∇u|2dx(3.26)0Ω

kC21≥0|∇u|2dx+Pk|u|2dx=Cu2,1+C21+C20C241PΩPPΩk0whereC4:=2>0.1+CPForthisestimateitisessentialthatusatisfiesthehomogeneousDirichletboundarycondition.(B)|c|>0:∞Firstofall,weconsiderasmoothfunctionu∈C∞(Ω).Fromu∇u=1∇u202wegetbyintegratingbyparts


1212c·∇uudx=c·∇udx=−∇·cudx.Ω2Ω2ΩSinceaccordingtoTheorem3.7thespaceC∞(Ω)isdenseinV,theabove0relationalsoholdsforu∈V.Consequently,byvirtueof(3.16)and(3.17)weobtain
12a(u,u)=K∇u·∇u+r−∇·cudxΩ2
(3.27)≥{k|∇u|2+r|u|2}dxforallu∈V.00ΩHence,adistinctionconcerningr0asin(A)withthesameresults(constants)ispossible.Summarizing,wehavethereforeproventhefollowingapplicationoftheLax–MilgramTheorem(Theorem3.1):Theorem3.12SupposeΩ⊂RdisaboundedLipschitzdomain.Undertheassumptions(3.15)–(3.17)thehomogeneousDirichletproblemhasoneandonlyoneweaksolutionu∈H1(Ω).0(II)MixedBoundaryConditions∂Ω=Γ,V=H1(Ω)2Supposeuisasolutionof(3.12),(3.19);thatis,intheclassicalsenseletu∈C2(Ω)∩C1(Ω)andthedifferentialequation(3.12)besatisfied¯pointwiseinΩand(3.19)pointwiseon∂Ωundertheassumptions(3.13),(3.21).However,theweakercasecanagainbeconsidered,nowundertheassumptions(3.14),(3.22),thatu∈H2(Ω)andthedifferentialequationissatisfiedinthesenseofL2(Ω)aswellastheboundarycondition(3.19)inthesenseofL2(∂Ω). 3.2.EllipticBoundaryValueProblems105Asin(I),accordingto(3.11),

a(u,v):={K∇u·∇v+c·∇uv+ruv}dx+αuvdσ(3.28)Ω

∂Ω=b(v):=fvdx+g2vdσforallv∈V.Ω∂ΩUndertheassumptions(3.15),(3.22)thecontinuityofbanda,respec-tively,((3.24)and(3.25))caneasilybeshown.Theadditionalnewtermscanbeestimated,forinstanceundertheassumptions(3.15),(3.22),bytheCauchy–SchwarzinequalityandtheTraceTheorem(Theorem3.4)asfollows:
g2vdσ≤g20,∂Ωv|∂Ω0,∂Ω≤Cg20,∂Ωv1forallv∈V∂Ωand
αuvdσ≤αu|v|≤C2αuv,∞,∂Ω∂Ω0,∂Ω∂Ω0,∂Ω∞,∂Ω11∂Ωrespectively,forallu,v∈V,whereC>0denotestheconstantappearingintheTraceTheorem.ConditionsfortheV-EllipticityofaFortheproofoftheV-ellipticityweproceedsimilarlyto(I)(B),butnowtakingintoaccountthemixedboundaryconditions.Fortheconvectivetermwehave



121212c·∇uudx=c·∇udx=−∇·cudx+ν·cudσ,Ω2Ω2Ω2∂Ωandthus

1212a(u,u)=K∇u·∇u+r−∇·cudx+α+ν·cudσ.Ω2∂Ω2Thisshowsthatα+1ν·c≥0on∂Ωshouldadditionallybeassumed.If2r0>0in(3.17),thentheV-ellipticityofafollowsdirectly.However,ifonlyr0≥0isvalid,thentheso-calledFriedrichs’inequality,arefinedversionofthePoincar´einequality,helps(see[25,Theorem1.9]).Theorem3.13SupposeΩ⊂RdisaboundedLipschitzdomainandletthesetΓ˜⊂∂Ωhaveapositive(d−1)-dimensionalmeasure.ThenthereexistssomeconstantC>0suchthatforallv∈H1(Ω),F

1/2v≤Cv2dσ+|∇v|2dx.(3.29)1FΓ˜ΩIfα+1ν·c≥α>0forx∈Γ˜⊂ΓandΓhasapositive(˜d−1)-202dimensionalmeasure,thenr0≥0isalreadysufficientfortheV-ellipticity. 1063.FiniteElementMethodsforLinearEllipticProblemsIndeed,usingTheorem3.13,wehave

a(u,u)≥k|u|2+αu2dσ≥min{k,α}|u|2+u2dσ≥Cu201000151Γ˜Γ˜−2withC5:=CFmin{k0,α0}.Therefore,weobtaintheexistenceanduniquenessofasolutionanalogouslytoTheorem3.12.(III)GeneralCaseFirst,weconsiderthecaseofahomogeneousDirichletboundaryconditiononΓ3with|Γ3|d−1>0.Forthis,wedefine1V:=v∈H(Ω):γ0(v)=0onΓ3.(3.30)HereVisaclosedsubspaceofH1(Ω),sincethetracemappingγ:0H1(Ω)→L2(∂Ω)andtherestrictionofafunctionfromL2(∂Ω)toL2(Γ)3arecontinuous.Supposeuisasolutionof(3.12),(3.18)–(3.20);thatis,inthesenseofclassicalsolutionsletu∈C2(Ω)∩C1(Ω)andthedifferentialequation¯(3.12)besatisfiedpointwiseinΩandtheboundaryconditions(3.18)–(3.20)pointwiseontheirrespectivepartsof∂Ωundertheassumptions(3.13),(3.21).However,theweakercasethatu∈H2(Ω)andthedifferentialequationissatisfiedinthesenseofL2(Ω)andtheboundaryconditions(3.18)–(3.20)aresatisfiedinthesenseofL2(Γ),j=1,2,3,underthejassumptions(3.14),(3.22)canalsobeconsideredhere.Asin(I),accordingto(3.11),

a(u,v):={K∇u·∇v+c·∇uv+ruv}dx+αuvdσ(3.31)ΩΓ2


=b(v):=fvdx+g1vdσ+g2vdσforallv∈V.ΩΓ1Γ2Undertheassumptions(3.15),(3.22)thecontinuityofaandb,(3.25))and((3.24)canbeprovenanalogouslyto(II).ConditionsforV-EllipticityofaFortheverificationoftheV-ellipticityweagainproceedsimilarlyto(II),butnowtheboundaryconditionsaremorecomplicated.Herewehavefortheconvectiveterm


1212c·∇uudx=−∇·cudx+ν·cudσ,Ω2Ω2Γ1∪Γ2andtherefore
12a(u,u)=K∇u·∇u+r−∇·cudxΩ2

1212+ν·cudσ+α+ν·cudσ.2Γ1Γ22 3.2.EllipticBoundaryValueProblems107InordertoensuretheV-ellipticityofaweneed,besidestheobviousconditions1ν·c≥0onΓ1andα+ν·c≥0onΓ2,(3.32)2thefollowingcorollaryfromTheorem3.13.Corollary3.14SupposeΩ⊂RdisaboundedLipschitzdomainandΓ˜⊂∂Ωhasapositive(d−1)-dimensionalmeasure.Thenthereexistssome>0suchthatforallv∈H1(Ω)withv|constantCFΓ˜=0,
1/2v≤C|∇v|2dx=C|v|.0FF1ΩThiscorollaryyieldsthesameresultsasinthecaseofhomogeneousDirichletboundaryconditionsonthewholeof∂Ω.If|Γ3|d−1=0,thenbytighteningconditions(3.32)forcandα,theapplicationofTheorem3.13asdonein(II)maybesuccessful.SummaryWewillnowpresentasummaryofourconsiderationsforthecaseofhomogeneousDirichletboundaryconditions.Theorem3.15SupposeΩ⊂RdisaboundedLipschitzdomain.Undertheassumptions(3.15),(3.16),(3.22)withg3=0,theboundaryvalueproblem(3.12),(3.18)–(3.20)hasoneandonlyoneweaksolutionu∈V,if(1)r−1∇·c≥0inΩ.2(2)ν·c≥0onΓ1.(3)α+1ν·c≥0onΓ.22(4)Additionally,oneofthefollowingconditionsissatisfied:(a)|Γ3|d−1>0.(b)ThereexistssomeΩ˜⊂Ωwith|Ω˜|d>0andr0>0suchthatr−1∇·c≥ronΩ˜.20(c)ThereexistssomeΓ˜1⊂Γ1with|Γ˜1|d−1>0andc0>0suchthatν·c≥c0onΓ˜1.(d)ThereexistssomeΓ˜2⊂Γ2with|Γ˜2|d−1>0andα0>0suchthatα+1ν·c≥αonΓ˜.202Remark3.16Wepointoutthatbyusingdifferenttechniquesintheproof,itispossibletoweakenconditions(4)(b)–(d)insuchawaythatonlythefollowinghastobeassumed:(b)x∈Ω:r−1∇·c>0>0,2d(c){x∈Γ1:ν·c>0}>0,d−1(d)x∈Γ:α+1ν·c>0>0.22d−1 1083.FiniteElementMethodsforLinearEllipticProblemsHowever,westressthattheconditionsofTheorem3.15areonlysuffi-cient,sinceconcerningtheV-ellipticity,itmightalsobepossibletobalanceanindefiniteaddendbysome“particulardefinite”addend.ButthiswouldrequireconditionsinwhichtheconstantsCPandCFareinvolved.NotethatthepureNeumannproblemforthePoissonequation−∆u=finΩ,(3.33)∂νu=gon∂ΩisexcludedbytheconditionsofTheorem3.15.Thisisconsistentwiththefactthatnotalwaysasolutionof(3.33)exists,andifasolutionexists,itobviouslyisnotunique(seeExercise3.8).BeforeweinvestigateinhomogeneousDirichletboundaryconditions,theapplicationofthetheoremwillbeillustratedbyanexampleofanaturalsituationdescribedinChapter0.Forthelinearstationarycaseofthedifferentialequation(0.33)intheform∇·(cu−K∇u)+˜ru=fweobtain,bydifferentiatingandrearrangingtheconvectiveterm,−∇·(K∇u)+c·∇u+(∇·c+˜r)u=f,whichgivestheform(3.12)withr:=∇·c+˜r.Theboundary∂ΩconsistsonlyoftwopartsΓ1andΓ2.Therein,Γ1anoutflowboundaryandΓ2aninflowboundary;thatis,theconditionsc·ν≥0onΓ1andc·ν≤0onΓ2hold.Frequentlyprescribedboundaryconditionsare−(cu−K∇u)·ν=−ν·cuonΓ1,−(cu−K∇u)·ν=g2onΓ2.Theyarebasedonthefollowingassumptions:OntheinflowboundaryΓ2thenormalcomponentofthetotal(mass)fluxisprescribedbutontheoutflowboundaryΓ1,onwhichintheextremecaseK=0theboundaryconditionswoulddropout,onlythefollowingisrequired:•thenormalcomponentofthetotal(mass)fluxiscontinuousoverΓ1,•theambientmassfluxthatisoutsideΩconsistsonlyofaconvectivepart,•theextensivevariable(forexample,theconcentration)iscontinuousoverΓ1,thatis,theambientconcentrationinxisalsoequaltou(x).Therefore,afteranobviousreformulationweget,inaccordancewiththedefinitionsofΓ1andΓ2dueto(3.18),(3.19),theNeumannboundary 3.2.EllipticBoundaryValueProblems109condition(3.18),andthemixedboundarycondition(3.19),K∇u·ν=0onΓ1,K∇u·ν+αu=g2onΓ2,whereα:=−ν·c.NowtheconditionsofTheorem3.15canbechecked:Wehaver−1∇·c=˜r+1∇·c;therefore,forthelattertermtheinequality22in(1)and(4)(b)mustbesatisfied.Further,theconditionν·c≥0onΓ1holdsduetothecharacterizationoftheoutflowboundary.Becauseofα+1ν·c=−1ν·c,thecondition(3)issatisfiedduetothedefinitionof22theinflowboundary.NowweaddressthecaseofinhomogeneousDirichletboundaryconditions(|Γ3|d−1>0).ThissituationcanbereducedtothecaseofhomogeneousDirich-letboundaryconditions,ifweareabletochoosesome(fixed)elementw∈H1(Ω)insuchawaythat(inthesenseoftrace)wehaveγ0(w)=g3onΓ3.(3.34)Theexistenceofsuchanelementwisanecessaryassumptionfortheexis-tenceofasolution˜u∈H1(Ω).Ontheotherhand,suchanelementwcanexistonlyifg3belongstotherangeofthemappingH1(Ω)v→γ(v)|∈L2(Γ).0Γ33However,thisisnotvalidforallg∈L2(Γ),sincetherangeofthetrace33operatorofH1(Ω)isapropersubsetofL2(∂Ω).Therefore,weassumetheexistenceofsuchanelementw.SinceonlythehomogeneityoftheDirichletboundaryconditionsofthetestfunctionsplaysaroleinderivation(3.31)ofthebilinearformaandthelinearformb,wefirstobtainwiththespaceV,definedin(3.30),andV˜:=v∈H1(Ω):γ(v)=gonΓ=v∈H1(Ω):v−w∈V033thefollowingvariationalformulation:Find˜u∈V˜suchthata(˜u,v)=b(v)forallv∈V.However,thisformulationdoesnotfitintothetheoreticalconceptofSection3.1sincethespaceV˜isnotalinearone.Ifweput˜u:=u+w,thenthisisequivalenttothefollowing:Findu∈Vsuchthata(u,v)=b(v)−a(w,v)=:˜b(v)forallv∈V.(3.35)NowwehaveavariationalformulationforthecaseofinhomogeneousDirichletboundaryconditionsthathastheformrequiredinthetheory. 1103.FiniteElementMethodsforLinearEllipticProblemsRemark3.17IntheexistenceresultofTheorem3.1,theonlyassumptionisthatbhastobeacontinuouslinearforminV.Ford=1andΩ=(a,b)thisisalsosatisfied,forinstance,forthespeciallinearformδ(v):=v(γ)forv∈H1(a,b),γwhereγ∈(a,b)isarbitrarybutfixed,sincebyLemma3.4thespaceH1(a,b)iscontinuouslyembeddedinthespaceC[a,b].Thus,ford=1pointsources(b=δγ)arealsoallowed.However,ford≥2thisdoesnotholdsinceH1(Ω)⊂C(Ω).¯Finally,wewillonceagainstatethegeneralassumptionsunderwhichthevariationalformulationoftheboundaryvalueproblem(3.12),(3.18)–(3.20)inthespace(3.30),V=v∈H1(Ω):γ(v)=0onΓ,03haspropertiesthatsatisfytheconditionsoftheLax–MilgramTheorem(Theorem3.1):•Ω⊂RdisaboundedLipschitzdomain.•k,c,∇·c,r∈L∞(Ω),f∈L2(Ω),i,j∈{1,...,d},and,ifiji|Γ∪Γ|>0,ν·c∈L∞(Γ∪Γ)(i.e.,(3.15)).12d−112•Thereexistssomeconstantk0>0suchthatinΩ,wehaveξ·K(x)ξ≥k|ξ|2forallξ∈Rd(i.e.,(3.16)),0•g∈L2(Γ),j=1,2,3,α∈L∞(Γ)(i.e.,(3.22)).jj2•Thefollowinghold:(1)r−1∇·c≥0inΩ.2(2)ν·c≥0onΓ1.(3)α+1ν·c≥0onΓ.22(4)Additionally,oneofthefollowingconditionsissatisfied:(a)|Γ3|d−1>0.(b)ThereexistssomeΩ˜⊂Ωwith|Ω˜|d>0andr0>0suchthatr−1∇·c≥ronΩ.˜20(c)ThereexistssomeΓ˜1⊂Γ1with|Γ˜1|d−1>0andc0>0suchthatν·c≥c0onΓ˜1.(d)ThereexistssomeΓ˜2⊂Γ2with|Γ˜2|d−1>0andα0>0suchthatα+1ν·c≥αonΓ˜.202•If|Γ|>0,thenthereexistssomew∈H1(Ω)withγ(w)=g3d−103onΓ3(i.e.,(3.34)). 3.2.EllipticBoundaryValueProblems1113.2.2AnExampleofaBoundaryValueProblemofFourthOrderTheDirichletproblemforthebiharmonicequationreadsasfollows:Findu∈C4(Ω)∩C1(Ω)suchthat¯$∆2u=finΩ,(3.36)∂νu=u=0on∂Ω,whered222∆u:=∆(∆u)=∂i∂ju.i,j=1Inthecased=1thiscollapsesto∆2u=u(4).Foru,v∈H2(Ω)itfollowsfromCorollary3.9that

(u∆v−∆uv)dx={u∂νv−∂νuv}dσΩ∂Ωandhenceforu∈H4(Ω),v∈H2(Ω)(byreplacinguwith∆uintheaboveequation),



2∆u∆vdx=∆uvdx−∂ν∆uvdσ+∆u∂νvdσ.ΩΩ∂Ω∂ΩForaLipschitzdomainΩwedefineH2(Ω):=v∈H2(Ω)v=∂v=0on∂Ω0νandobtainthevariationalformulationof(3.36)inthespaceV:=H2(Ω):0Findu∈V,suchthat

a(u,v):=∆u∆vdx=b(v):=fvdxforallv∈V.ΩΩMoregeneral,foraboundaryvalueproblemoforder2minconservativeform,weobtainavariationalformulationinHm(Ω)orHm(Ω).03.2.3RegularityofBoundaryValueProblemsInSection3.2.1westatedconditionsunderwhichthelinearellipticbound-aryvalueproblemadmitsauniquesolutionu(˜u,respectively)insomesubspaceVofH1(Ω).Inmanycases,forinstancefortheinterpolationofthesolutionorinthecontextoferrorestimates(alsoinnormsotherthanthe·Vnorm)itisnotsufficientthatu(˜u,respectively)haveonlyfirstweakderivativesinL2(Ω).Therefore,withintheframeworkoftheso-calledregularitytheory,thequestionoftheassumptionsunderwhichtheweaksolutionbelongstoH2(Ω),forinstance,hastobeanswered.Theseadditionalconditionscontainconditionsabout 1123.FiniteElementMethodsforLinearEllipticProblems•thesmoothnessoftheboundaryofthedomain,•theshapeofthedomain,•thesmoothnessofthecoefficientsandtheright-handsideofthedifferentialequationandtheboundaryconditions,•thekindofthetransitionofboundaryconditionsinthosepoints,wherethetypeischanging,whichcanbequiterestrictiveasawhole.Therefore,inwhatfollowsweoftenassumeonlytherequiredsmoothness.Hereweciteasanexampleoneregularityresult([13,Theorem8.12]).Theorem3.18SupposeΩisaboundedC2-domainandΓ=∂Ω.Further,3assumethatk∈C1(Ω)¯,c,r∈L∞(Ω),f∈L2(Ω),i,j∈{1,...,d},ijiaswellas(3.16).Supposethereexistssomefunctionw∈H2(Ω)withγ0(w)=g3onΓ3.Letu˜=u+wandletubeasolutionof(3.35).Thenu˜∈H2(Ω)andu˜2≤C{u0+f0+w2}withaconstantC>0independentofu,f,andw.Onedrawbackoftheaboveresultisthatitexcludespolyhedraldomains.IftheconvexityofΩisadditionallyassumed,thenitcanbetransferredtothiscase.Simpleexamplesofboundaryvalueproblemsindomainswithreentrantcornersshowthatonecannotavoidsuchadditionalassumptions(seeExercise3.5).Exercises3.5Considertheboundaryvalueproblem(1.1),(1.2)forf=0inthesectorΩ:=(x,y)∈R2x=rcosϕ,y=rsinϕwith0C1.Inordertobeabletomakeacomparisonbetweenthevariants(a)and(b),weconsiderinthefollowingthecaseofarectangleΩ=(0,a)×(0,b).Thenumberofthenodesisthenproportionalto1/h2iftheelementsareall“essentially”ofthesamesize.However,ifweconsiderthenumberofnodes√Masgiven,thenhisproportionalto1/M.Usingthisintheestimate(3.67),wegetforasolutionu∈H2(Ω),1inthecase(a)forh/2:u−uh/21≤C1√,2M1inthecase(b)forh:u−uh1≤C¯1√.MIfbothconstantsarethesame,thismeansanadvantageforthevariant(a).Ontheotherhand,ifthesolutionissmootherandsatisfiesu∈H3(Ω),thentheestimate(3.68),whichcanbeappliedonlytothevariant(b),yields1inthecase(a)forh/2:u−uh/21≤C1√,2M1inthecase(b)forh:u−uh1≤C2.MByanelementaryreformulation,weget11C2√2C2<(<)C1⇐⇒M>(>)42,M2MC1whichgivesanadvantagefor(b)ifthenumberofvariablesMischosen,dependingonC2/C1,sufficientlylarge.However,thedenserpopulationofthematrixin(b)hastobeconfrontedwiththis.Hence,ahigher-orderpolynomialansatzhasanadvantageonlyifthesmoothnessofthesolutionleadstoahigherconvergencerate.Especiallyfornonlinearproblemswithless-smoothsolutions,apossibleadvantageofthehigher-orderansatzhastobeexaminedcritically. 1303.FiniteElementMethodsforLinearEllipticProblemsExercises3.10Provetheimplication“⇒”inTheorem3.20.Hint:Forv∈Vhdefineafunctionwibywi|int(K):=∂iv,i=1,...,d,andshowthatwiistheithpartialderivativeofv.3.11ConstructtheelementstiffnessmatrixforthePoissonequationonarectanglewithquadraticbilinearrectangularelements.VerifythatthisfiniteelementdiscretizationoftheLaplaceoperatorcanbeinterpretedasafinitedifferencemethodwiththedifferencestencilaccordingto(1.22).3.12Provethat:(a)dimP(Rd)=d+k.kk(b)P(Rd)|=P(K)ifint(K)=∅.kKk3.13Proveforgivenvectorsa,...,a∈Rdthata−a,...,a−1d+121d+1a1arelinearindependentifandonlyifa1−ai,...,ai−1−ai,ai+1−ai,...,ad+1−aiarelinearlyindependentforsomei∈{2,...,d}.3.14Determineforthepolynomialansatzonthecuboidasreferenceelement(3.59)theansatzspacePthatisobtainedbyanaffine-lineartransformationtoad-epiped.3.15SupposeKisarectanglewiththe(counterclockwisenumbered)ver-ticesa1,...,a4andthecorrespondingedgemidpointsa12,a23,a34,a41.ShowthattheelementsfofQ1(K)arenotdetermineduniquelybythedegreesoffreedomf(a12),f(a23),f(a34),f(a41).3.16Checkthegivenshapefunctionsfor(3.55)and(3.56).3.17DefineareferenceelementinR3by010Kˆ=conv{aˆ1,aˆ2,aˆ3}×[0,1]withˆa1=,aˆ2=,aˆ3=,001Pˆ=p(x,x)p(x)p∈P(R2),p∈P(R),112231121Σ=ˆp(ˆx)xˆ=(ˆai,j),i=0,1,2,j=0,1.Showtheuniquesolvabilityofthelocalinterpolationproblemanddescribetheelementsobtainedbyaffine-lineartransformation.3.18Supposed+1pointsa,j=1,...,d+1,inRdaregivenwiththejpropertyasinExercise3.13.Additionally,wedefineasin(3.48),(3.49)thebarycentriccoordinatesλj=λj(x;S)ofxwithrespecttothed-simplexSgeneratedbythepointsaj.Showthatforeachbijectiveaffine-linear 3.4.ConvergenceRateEstimates131mapping:Rd→Rd,λ(x;S)=λ((x);(S)),whichmeansthatthejjbarycentriccoordinatesareinvariantundersuchtransformations.3.19DiscussforthecubicHermiteansatz(3.64)andDirichletboundaryconditionsthechoiceofthedegreesoffreedomwithregardtotheanglebetweentwoedgesofboundaryelementsthatiseitherα=2πorα=2π.3.20ConstructanodalbasisfortheBogner–Fox–Schmitelementin(3.65).3.4ConvergenceRateEstimatesInthissectionweconsiderfurtherafiniteelementapproximationintheframeworkdescribedintheprevioussection:TheboundedbasicdomainΩ⊂Rdoftheboundaryvalueproblemisdecomposedintoconformingtri-angulationsTh,whichmayalsoconsistofdifferenttypesofelements.Here,byanelementwemeannotonlythesetK∈Th,butthisequippedwithsomeansatzspacePKanddegreesoffreedomΣK.However,theelementsaresupposedtodecomposeintoafixednumberofsubsets,independentofh,eachconsistingofelementsthatareaffineequivalenttoeachother.DifferentelementshavetobecompatiblewitheachothersuchthattheansatzspaceVh,introducedin(3.62),iswell-defined.Thesmoothnessofthefunctionsarisinginthiswayhastobeconsistentwiththeboundaryvalueproblem,insofarasVh⊂Visguaranteed.Inthefollowingweconsideronlyoneelementtype;thegeneralizationtothemoregeneralsit-uationwillbeobvious.Thegoalistoproveaprioriestimatesoftheformu−u≤C|u|hα(3.69)hwithconstantsC>0,α>0andnormsandseminorms·and|·|,respectively.WedonotattempttogivetheconstantCexplicitly,althoughinprin-ciple,thisispossible(withothertechniquesofproof).Inparticular,inthefollowingChastobeunderstoodgenerically;thatis,byCwedenoteatdifferentplacesdifferentvalues,which,however,areindependentofh.Therefore,theestimate(3.69)doesnotserveonlytoestimatenumericallytheerrorforafixedtriangulationTh.Itisratherusefulforestimatingwhatgaininaccuracycanbeexpectedbyincreasingtheeffort,whichthencorre-spondstothereductionofhbysomerefinement(seethediscussionaround(3.67)).Independentlyoftheconvergencerateα,(3.69)providesthecer-taintythatanarbitraryaccuracyinthedesirednorm·canbeobtainedatall.Inthefollowing,wewillimposesomegeometricconditionsonthefamily(Th)h,whichhavealwaystobeunderstooduniformlyinh.Forafixedtriangulationtheseconditionsarealwaystriviallysatisfied,sincehere 1323.FiniteElementMethodsforLinearEllipticProblemswehaveafinitenumberofelements.Forafamily(Th)hwithh→0,thusforincreasingrefinement,thisnumberbecomesunbounded.Inthefollowingestimateswehavethereforetodistinguishbetween“variable”valueslikethenumberofnodesM=M(h)ofTh,and“fixed”valueslikethedimen-siondorthedimensionofPKorequivalenceconstantsintherenormingofPK,whichcanallbeincludedinthegenericconstantC.3.4.1EnergyNormEstimatesIfwewanttoderiveestimatesinthenormoftheHilbertspaceVunderlyingthevariationalequationfortheboundaryvalueproblem,concretely,inthenormofSobolevspaces,thenC´ea’slemma(Theorem2.17)showsthatforthispurposeitisnecessaryonlytospecifyacomparisonelementvh∈Vhforwhichtheinequalityu−v≤C|u|hα(3.70)hholds.For·=·1,theseestimatesarecalledenergynormestimatesduetotheequivalenceof·1and·a(cf.(2.46))inthesymmetriccase.Therefore,thecomparisonelementvhhastoapproximateuaswellaspossible,andingenera,litisspecifiedastheimageofalinearoperatorIh:vh=Ih(u).TheclassicalapproachconsistsinchoosingforIhtheinterpolationoper-atorwithrespecttothedegreesoffreedom.Tosimplifythenotation,werestrictourselvesinthefollowingtoLagrangeelements,thegeneralizationtoHermiteelementsisalsoeasilypossible.WesupposethatthetriangulationThhasitsdegreesoffreedominthenodesa1,...,aMwiththecorrespondingnodalbasisϕ1,...,ϕM.ThenletMIh(u):=u(ai)ϕi∈Vh.(3.71)i=1ForthesakeofIh(u)beingwell-defined,u∈C(Ω)hastobeassumedin¯ordertoensurethatucanbeevaluatedinthenodes.Thisrequiresacertainsmoothnessassumptionaboutthesolutionu,whichweformulateasu∈Hk+1(Ω).Thus,ifweassumeagaind≤3forthesakeofsimplicity,theembeddingtheorem(Theorem3.10)ensuresthatIiswell-definedonHk+1(Ω)forhk≥1.FortheconsideredC0-elements,wehaveI(u)∈H1(Ω)byvirtuehofTheorem3.20.Therefore,wecansubstantiatethedesiredestimate(3.70)toαu−Ih(u)1≤Ch|u|k+1.(3.72) 3.4.ConvergenceRateEstimates133Sobolev(semi)normscanbedecomposedintoexpressionsoversubsetsofΩ,thus,forinstance,theelementsofTh,

|u|2=|∂αu|2dx=|∂αu|2dx=|u|2,ll,KΩK|α|=lK∈Th|α|=lK∈Thand,correspondingly,u2=u2,ll,KK∈Thwhere,ifΩisnotbasicdomain,thiswillbeincludedintheindicesofthenorm.SincetheelementsKareconsideredasbeingclosed,Kshouldmorepreciselybereplacedbyint(K).Byvirtueofthisdecomposition,itissufficienttoprovetheestimate(3.72)fortheelementsK.Thishassomeanalogytothe(elementwise)assemblingdescribedinSection2.4.2,whichisalsotobeseeninthefollowing.OnK,theoperatorIhreducestotheanalogouslydefinedlocalinterpolationoperator.SupposethenodesofthedegreesoffreedomonKareai1,...,aiL,whereL∈NisthesameforallK∈Thduetotheequivalenceofelements.ThenIh(u)|K=IK(u|K)foru∈C(Ω)¯,whereLIK(u):=u(aij)ϕijforu∈C(K),j=1sincebothfunctionsofPKsolvethesameinterpolationproblemonK(cf.Lemma2.10).Sincewehavean(affine)equivalenttriangulation,theproofofthelocalestimateαu−IK(u)m,K≤Ch|u|k+1,K(3.73)isgenerallydoneinthreesteps:•TransformationtosomereferenceelementKˆ,•Proofof(3.73)onKˆ,•BacktransformationtotheelementK.Tobeprecise,theestimate(3.73)willevenbeprovedwithhKinsteadofh,wherehK:=diam(K)forK∈Th,andinthesecondstep,thefixedvaluehKˆisincorporatedintheconstant.ThepowersofhKareduetothetransformationsteps.Therefore,letsomereferenceelementKˆwiththenodesˆa1,...,aˆLbechosenasfixed.Byassumption,thereexistssomebijective,affine-linear 1343.FiniteElementMethodsforLinearEllipticProblemsmappingF=FK:Kˆ→K,(3.74)F(ˆx)=Bxˆ+d,(cf.(2.30)and(3.57)).Bythistransformation,functionsv:K→Raremappedtofunctionsˆv:Kˆ→Rbyvˆ(ˆx):=v(F(ˆx)).(3.75)Thistransformationisalsocompatiblewiththelocalinterpolationoperatorinthefollowingsense:I
K(v)=IKˆ(ˆv)forv∈C(K).(3.76)ThisfollowsfromthefactthatthenodesoftheelementsaswellastheshapefunctionsaremappedontoeachotherbyF.Foraclassicallydifferentiablefunctionthechainrule(see(2.49))implies∇v(F(ˆx))=B−T∇vˆ(ˆx),(3.77)xxˆandcorrespondingformulasforhigher-orderderivatives,forinstance,2−T2−1Dxv(F(ˆx))=BDxˆvˆ(ˆx)B,whereD2v(x)denotesthematrixofthesecond-orderderivatives.Thesexchainrulesholdalsoforcorrespondingv∈Hl(K)(Exercise3.22).Thesituationbecomesparticularlysimpleinonespacedimension(d=1).Theconsideredelementsreducetoapolynomialansatzonsimplices,whichhereareintervals.ThusF:Kˆ=[0,1]→K=[ai1,ai2],xˆ→hKxˆ+ai1,wherehK:=ai2−ai1denotesthelengthoftheelement.Hence,forl∈N,∂lv(F(ˆx))=h−l∂lvˆ(ˆx).xKxˆBythesubstitutionruleforintegrals(cf.(2.50))anadditionalfactor|det(B)|=harisessuchthat,forv∈Hl(K),wehaveK2l−1212|v|l,K=|vˆ|.hl,KˆKHence,for0≤m≤k+1itfollowsby(3.76)that2m−1|v−I(v)|2=1vˆ−I(ˆv)2.Km,KhKˆm,KˆKThus,whatismissing,isanestimateofthetypevˆ−IKˆ(ˆv)≤C|vˆ|k+1,Kˆ(3.78)m,Kˆ 3.4.ConvergenceRateEstimates135forˆv∈Hk+1(Kˆ).Inspecificcasesthiscanpartlybeprovendirectlybutinthefollowingageneralproof,whichisalsoindependentofd=1,willbesketched.Forthis,themappingG:Hk+1(Kˆ)→Hm(Kˆ),(3.79)vˆ→vˆ−IKˆ(ˆv),isconsidered.Themappingislinearbutalsocontinuous,sinceLI(ˆv)≤vˆ(ˆa)ˆϕKˆm,Kˆiii=1k+1,Kˆ(3.80)L≤ϕˆik+1,Kˆvˆ∞,Kˆ≤Cvˆk+1,Kˆ,i=1wherethecontinuityoftheembeddingofHk+1(Kˆ)inHm(Kˆ)(see(3.8))andofHk+1(Kˆ)inC(Kˆ)(Theorem3.10)isused,andthenormcontributionfromthefixedbasisfunctionsˆϕiisincludedintheconstant.IftheansatzspacePˆischoseninsuchawaythatPk⊂Pˆ,thenGhastheadditionalpropertyG(p)=0forp∈Pk,sincethesepolynomialsareinterpolatedthenexactly.Suchmappingssat-isfytheBramble–Hilbertlemma,whichwilldirectlybeformulated,forfurtheruse,inamoregeneralway.Theorem3.24(Bramble–Hilbertlemma)SupposeK⊂Rdisopen,k∈N,1≤p≤∞,andG:Wk+1(K)→Risa0pcontinuouslinearfunctionalthatsatisfiesG(q)=0forallq∈Pk.(3.81)ThenthereexistssomeconstantC>0independentofGsuchthatforallv∈Wk+1(K)p|G(v)|≤CG|v|k+1,p,K.Proof:See[9,Theorem28.1].
HereGdenotestheoperatornormofG(see(A4.25)).Theestimatewiththefullnorm·k+1,p,Kontheright-handside(andC=1)wouldhenceonlybetheoperatornorm’sdefinition.Thecondition(3.81)allowsthereductiontothehighestseminorm.FortheapplicationoftheBramble–Hilbertlemma(Theorem3.24),whichwasformulatedonlyforfunctionals,totheoperatorGaccordingto(3.79)anadditionalargumentisrequired(alternatively,Theorem3.24couldbegeneralized): 1363.FiniteElementMethodsforLinearEllipticProblemsGenerally,forˆw∈Hm(Kˆ)(asineverynormedspace)wehavewˆ=supϕ(ˆw),(3.82)m,Kˆϕ∈(Hm(Kˆ))ϕ≤1wherethenormapplyingtoϕistheoperatornormdefinedin(A4.25).Foranyfixedϕ∈(Hm(Kˆ))
thelinearfunctionalonHk+1(Kˆ)isdefinedbyG˜(ˆv):=ϕ(G(ˆv))forvˆ∈Hk+1(Kˆ).(3.83)Accordingto(3.80),G˜iscontinuousanditfollowsthatG˜≤ϕG.Theorem3.24isapplicabletoG˜andyields|G˜(ˆv)|≤CϕG|vˆ|.k+1,KˆBymeansof(3.82)itfollowsthatG(ˆv)≤CG|vˆ|.m,Kˆk+1,KˆThesameproofcanalsobeusedintheproofofTheorem3.31(3.94).AppliedtoGdefinedin(3.79),theestimate(3.80)showsthattheoperatornormId−IKˆcanbeestimatedindependentlyfromm(butdependentonkandtheˆϕi)andcanbeincorporatedintheconstantthatgives(3.78)ingeneral,independentoftheone-dimensionalcase.Therefore,intheone-dimensionalcasewecancontinuewiththeestimationandget2m−12121−2m+2(k+1)−12|v−IK(v)|m,K≤hC|vˆ|k+1,Kˆ≤C(hK)|v|k+1,K.KSinceduetoI(v)∈H1(Ω)wehaveform=0,1h|v−I(v)|2=|v−I(v)|2,Km,KhmK∈ThwehaveproventhefollowingTheorem:Theorem3.25ConsiderinonespacedimensionΩ=(a,b)thepolyno-mialLagrangeansatzonelementswithmaximumlengthhandsupposethatfortherespectivelocalansatzspacesP,theinclusionPk⊂Pissatisfiedforsomek∈N.ThenthereexistssomeconstantC>0suchthatforallv∈Hk+1(Ω)and0≤m≤k+1,1/2|v−I(v)|2≤Chk+1−m|v|.Km,Kk+1K∈ThIfthesolutionuoftheboundaryvalueproblem(3.12),(3.18)–(3.20)belongstoHk+1(Ω),thenwehaveforthefiniteelementapproximationuaccordingh 3.4.ConvergenceRateEstimates137to(3.39),ku−uh1≤Ch|u|k+1.Notethatford=1adirectproofisalsopossible(seeExercise3.21).Nowweaddresstothegenerald-dimensionalsituation:Theseminorm|·|1istransformed,forinstance,asfollows(cf.(2.49)):

|v|2=|∇v|2dx=B−T∇vˆ·B−T∇vˆ|det(B)|dx.ˆ(3.84)1,KxxˆxˆKKˆFromthis,itfollowsforˆv∈H1(Kˆ)that|v|≤CB−1|det(B)|1/2|vˆ|.1,K1,KˆSincedisoneofthementioned“fixed”quantitiesandallnormsonRd,dareequivalent,thematrixnorm·canbechosenarbitrarily,anditisalsopossibletochangebetweensuchnorms.IntheaboveconsiderationsKandKˆhadequalrights;thussimilarlyforv∈H1(K),wehave−1/2|vˆ|1,Kˆ≤CB|det(B)||v|1,K.Ingeneral,wehavethefollowingtheorem:Theorem3.26SupposeKandKˆareboundeddomainsinRdthataremappedontoeachotherbyanaffinebijectivelinearmappingF,definedin(3.74).Ifv∈Wl(K)forl∈Nandp∈[1,∞],thenwehaveforvˆ(definedpin(3.75)),vˆ∈Wl(Kˆ),andforsomeconstantC>0independentofv,p|vˆ|≤CBl|det(B)|−1/p|v|,(3.85)l,p,Kˆl,p,K|v|≤CB−1l|det(B)|1/p|vˆ|.(3.86)l,p,Kl,p,KˆProof:See[9,Theorem15.1].
Forfurtheruse,alsothistheoremhasbeenformulatedinamoregeneralwaythanwouldbenecessaryhere.Here,onlythecasep=2isrelevant.Hence,ifweusetheestimateofTheorem3.24,thenthevalueB(forsomematrixnorm)hastoberelatedtothegeometryofK.Forthis,letforK∈Th,:=supdiam(S)SisaballinRdandS⊂K.KHence,inthecaseofatriangle,hKdenotesthelongestedgeandKthediameteroftheinscribedcircle.Similarly,thereferenceelementhasits(fixed)parametershˆandˆ.Forexample,forthereferencetrianglewiththeverticesˆa=(0,0),ˆa=(1,0),ˆa=(0,1)wehavethathˆ=21/2and123ˆ=2−21/2. 1383.FiniteElementMethodsforLinearEllipticProblemsTheorem3.27ForF=FKaccordingto(3.74),inthespectralnorm·2,wehavehK−1ˆhB2≤andB2≤.ˆKProof:SinceKandKˆhaveequalrightsintheassertion,itsufficestoproveoneofthestatements:Wehave(cf.(A4.25))11B2=supBξ=sup|Bξ|2.|ξ|2=ˆˆ2ˆ|ξ|2=ˆForeveryξ∈Rdwith|ξ|=ˆthereexistsomepointsˆy,zˆ∈Kˆsuchthat2yˆ−zˆ=ξ.SinceBξ=F(ˆy)−F(ˆz)andF(ˆy),F(ˆz)∈K,wehave|Bξ|2≤hK.Consequently,bytheaboveidentitywegetthefirstinequality.
Ifwecombinethelocalestimatesof(3.78),Theorem3.26,andTheorem3.27,weobtainforv∈Hk+1(K)and0≤m≤k+1,mhKk+1−m|v−IK(v)|m,K≤ChK|v|k+1,K,(3.87)KwhereˆandˆhareincludedintheconstantC.Inordertoobtainsomeconvergencerateresult,wehavetocontrolthetermhK/K.Ifthistermisbounded(uniformlyforalltriangulations),wegetthesameestimateasintheone-dimensionalcase(whereevenhK/K=1).Conditionsoftheform1+αK≥σhKforsomeσ>0and0≤α0suchthatforallh>0andallK∈Th,K≥σhK.Fromestimate(3.87)weconcludedirectlythefollowingtheorem:Theorem3.29ConsiderafamilyofLagrangefiniteelementdiscretiza-tionsinRdford≤3onaregularfamilyoftriangulations(T)inthehhgeneralitydescribedattheverybeginning.FortherespectivelocalansatzspacesPsupposePk⊂Pforsomek∈N.ThenthereexistssomeconstantC>0suchthatforallv∈Hk+1(Ω)and0≤m≤k+1,1/2|v−I(v)|2≤Chk+1−m|v|.(3.88)Km,Kk+1K∈Th 3.4.ConvergenceRateEstimates139Ifthesolutionuoftheboundaryvalueproblem(3.12),(3.18)–(3.20)belongstoHk+1(Ω),thenforthefiniteelementapproximationudefinedin(3.39),hitfollowsthatu−u≤Chk|u|.(3.89)h1k+1Remark3.30Indeed,hereandalsoinTheorem3.25asharperestimatehasbeenshown,which,forinstancefor(3.89),hasthefollowingform:1/2u−u≤Ch2k|u|2.(3.90)h1Kk+1,KK∈ThInthefollowingwewilldiscusswhattheregularityassumptionmeansinthetwosimplestcases:ForarectangleandthecuboidK,whoseedgelengthscanbeassumed,withoutanylossofgenerality,tobeoforderh1≤h2[≤h3],wehave20211/2hKh2h3=1++.Kh1h1Thistermisuniformlyboundedifandonlyifthereexistssomeconstantα(≥1)suchthath1≤h2≤αh1,(3.91)h1≤h3≤αh1.Inordertosatisfythiscondition,arefinementinonespacedirectionhastoimplyacorrespondingoneintheotherdirections,althoughincertainanisotropicsituationsonlytherefinementinonespacedirectionisrecom-mendable.If,forinstance,theboundaryvalueproblem(3.12),(3.18)–(3.20)withc=r=0,butspace-dependentconductivityK,isinterpretedasthesimplestgroundwatermodel(see(0.18)),thenitistypicalthatKvariesdiscontinuouslyduetosomelayeringormorecomplexgeologicalstructures(seeFigure3.11).K1K2K1Figure3.11.Layeringandanisotropictriangulation.Ifthinlayersariseinsuchacase,ontheonehandtheyhavetoberesolved;thatis,thetriangulationhastobecompatiblewiththelayeringandthere 1403.FiniteElementMethodsforLinearEllipticProblemshavetobesufficientlymanyelementsinthislayer.Ontheotherhand,thesolutionoftenchangeslessstronglyinthedirectionofthelayeringthanovertheboundariesofthelayer,whichsuggestsananisotropictriangulation,thatis,astronglyvaryingdimensioningoftheelements.Therestriction(3.91)isnotcompatiblewiththis,butinthecaseofrectanglesthisisdueonlytothetechniquesofproof.Inthissimplesituation,thelocalinterpolationerrorestimatecanbeperformeddirectly,atleastforP=Q1(K),withoutanytransformationsuchthattheestimate(3.89)(fork=1)isobtainedwithoutanyrestrictionslike(3.91).ThenextsimpleexampleisatriangleK:Thesmallestangleαmin=αmin(K)includesthelongestedgehK,andwithoutlossofgenerality,thesituationisasillustratedinFigure3.12.a3a1αminh2hKa2Figure3.12.Trianglewiththelongestedgeandtheheightasparameters.Forthe2×2matrixB=(a2−a1,a3−a1),intheFrobeniusnorm·F(see(A3.5))wehave−11BF=BF,|det(B)|andfurther,withtheheighth2overhK,det(B)=hKh2,(3.92)sincedet(B)/2istheareaofthetriangle,aswellasB2=|a−a|2+|a−a|2≥h2,F212312KsuchthatBB−1≥h/h,FFK2andthusbyvirtueofcotαmincotα.FFminSincewegetbyanalogousestimatesBB−1≤4cotα,FFminitfollowsthatcotαdescribestheasymptoticbehaviorofBB−1forminafixedchosenarbitrarymatrixnorm.Therefore,fromTheorem3.27we 3.4.ConvergenceRateEstimates141gettheexistenceofsomeconstantC>0independentofhsuchthatforallK∈Th,hK≥Ccotαmin(K).(3.93)KConsequently,afamilyoftriangulations(Th)hoftrianglescanonlybereg-ularifallanglesofthetrianglesareuniformlyboundedfrombelowbysomepositiveconstant.Thisconditionsometimesiscalledtheminimumanglecondition.InthesituationofFigure3.11itwouldthusnotbeal-lowedtodecomposetheflatrectanglesinthethinlayerbymeansofaFriedrichs–Kellertriangulation.Obviously,usingdirectlytheestimatesofTheorem3.26weseethattheminimumangleconditionissufficientfortheestimatesofTheorem3.29.Thisstillleavesthepossibilityopenthatlesssevereconditionsarealsosufficient.3.4.2TheMaximumAngleConditiononTrianglesInwhatfollowsweshowthatthecondition(3.93)isdueonlytothetech-niquesofproof,andatleastinthecaseofthelinearansatz,ithasindeedonlytobeenssuredthatthelargestangleisuniformlyboundedawayfromπ.Therefore,thisallowstheapplicationofthedescribedapproachinthelayerexampleofFigure3.11.Theestimate(3.87)showsthatform=0thecrucialpartdoesnotarise;henceonlyform=k=1dotheestimateshavetobeinvestigated.Itturnsouttobeusefultoprovethefollowingsharperformoftheestimate(3.78):Theorem3.31ForthereferencetriangleKˆwithlinearansatzfunctionsthereexistssomeconstantC>0suchthatforallvˆ∈H2(Kˆ)andj=1,2,∂∂vˆ−IKˆ(ˆv)≤Cvˆ.∂xˆj0,Kˆ∂xˆj1,KˆProof:Inordertosimplifythenotation,wedropthehatˆinthenotationofthereferencesituationintheproof.Hence,wehaveK=conv{a1,a2,a3}witha=(0,0)T,a=(1,0)T,anda=(0,1)T.Weconsiderthefollowing123linearmappings:F:H1(K)→L2(K)isdefinedby1
1F1(w):=w(s,0)ds,0and,analogously,F2astheintegralovertheboundarypartconv{a1,a3}.TheimageistakenasconstantfunctiononK.ByvirtueoftheTraceThe-orem(Theorem3.5),andthecontinuousembeddingofL2(0,1)inL1(0,1),theFiarewell-definedandcontinuous.Sincewehaveforw∈P0(K),Fi(w)=w, 1423.FiniteElementMethodsforLinearEllipticProblemstheBramble–Hilbertlemma(Theorem3.24)impliestheexistenceofsomeconstantC>0suchthatforw∈H1(K),Fi(w)−w0,K≤C|w|1,K.(3.94)Thiscanbeseeninthefollowingway:Letv∈H1(K)bearbitrarybutfixed,andforthis,consideronH1(K)thefunctionalG(w):=F(w)−w,F(v)−vforw∈H1(K).iiWehaveG(w)=0forw∈P0(K)and|G(w)|≤Fi(w)−w0,KFi(v)−v0,K≤CFi(v)−v0,Kw1,Kbytheaboveconsideration.ThusbyTheorem3.24,|G(w)|≤CFi(v)−v0,K|w|1,K.Forv=wthisimplies(3.94).Ontheotherhand,forw:=∂1vitfollowsthatF1(∂1v)=v(1,0)−v(0,0)=(IK(v))(1,0)−(IK(v))(0,0)==∂1(IK(v))(x1,x2)for(x1,x2)∈Kand,analogously,F2(∂2v)=∂2(IK(v))(x1,x2).This,substitutedinto(3.94),givestheassertion.
Comparedwithestimate(3.78),forexampleinthecasej=1theterm2∂vˆdoesnotariseontheright-handside:Thederivativesandthusthe∂xˆ22spacedirectionsarethereforetreated“moreseparately.”Next,theeffectofthetransformationwillbeestimatedmoreprecisely.Forthis,letαmax=αmax(K)bethelargestanglearisinginK∈Th,supposedtoincludethevertexa1,andleth1=h1K:=|a2−a1|2,h2=h2K:=|a3−a1|(seeFigure3.13).a1hαmax2h1a3a2Figure3.13.Ageneraltriangle.Asavariantof(3.86)(forl=1)wehavethefollowing: 3.4.ConvergenceRateEstimates143Theorem3.32SupposeKisageneraltriangle.Withtheabovenotationforv∈H1(K)andthetransformedvˆ∈H1(Kˆ),221/2√∂∂|v|≤2|det(B)|−1/2h2vˆ+h2vˆ.1,K2∂xˆ1∂xˆ10,Kˆ20,KˆProof:Wehaveb11b12B=(a2−a1,a3−a1)=:b21b22andhenceb11b12b=h1,b=h2.(3.95)2122From1b22−b21−TB=det(B)−b12b11and(3.84)itthusfollowsthat
221b22∂−b21∂|v|1,K=vˆ+vˆdxˆ|det(B)|Kˆ−b12∂xˆ1b11∂xˆ2andfromthistheassertion.
Inmodificationoftheestimate(3.85)(forl=2)weprovethefollowingresult:Theorem3.33SupposeKisageneraltrianglewithdiameterhK=diam(K).Withtheabovenotationforvˆ∈H2(Kˆ)andthetransformedv∈H2(K),∂vˆ≤4|det(B)|−1/2hh|v|fori=1,2.∂xˆiK2,Ki1,KˆProof:Accordingto(3.84)wegetbyexchangingKandKˆ,
|wˆ|2=BT∇w·BT∇wdx|det(B)|−11,KˆxxKand,consequently,forˆw=∂vˆ,thusby(3.77)forw=(BT∇v),∂xˆixi2
∂2vˆ=BT∇BT∇vdx|det(B)|−1.∂xˆxxii1,KˆKAccordingto(3.95),thenormoftheithrowvectorofBTisequaltoh,iwhichimpliestheassertion.
1443.FiniteElementMethodsforLinearEllipticProblemsInsteadoftheregularityofthefamilyoftriangulationsandhencetheuniformboundforcotαmin(K)(see(3.93))werequirethefollowingdefinition:Definition3.34Afamilyoftriangulations(Th)hoftrianglessatisfiesthemaximumangleconditionifthereexistssomeconstantα<πsuchthatforallh>0andK∈Ththemaximumangleαmax(K)ofKsatisfiesαmax(K)≤α.Sinceαmax(K)≥π/3isalwayssatisfied,themaximumangleconditionisequivalenttotheexistenceofsomeconstant˜s>0,suchthatsin(αmax(K))≥s˜forallK∈Thandh>0.(3.96)Therelationofthisconditiontotheaboveestimatesisgivenby(cf.(3.92))det(B)=h1h2sinαmax.(3.97)InsertingtheestimatesofTheorem3.32(forv−IK(v)),Theorem3.31,andTheorem3.33intoeachotherandrecalling(3.96),(3.97),thefollowingtheoremfollowsfromC´ea’slemma(Theorem2.17):Theorem3.35Considerthelinearansatz(3.53)onafamilyoftriangu-lations(Th)hoftrianglesthatsatisfiesthemaximumanglecondition.ThenthereexistssomeconstantC>0suchthatforv∈H2(Ω),v−Ih(v)1≤Ch|v|2.Ifthesolutionuoftheboundaryvalueproblem(3.12),(3.18)–(3.20)belongstoH2(Ω),thenforthefiniteelementapproximationudefinedin(3.39)hwehavetheestimateu−uh1≤Ch|u|2.(3.98)Exercise3.26showsthenecessityofthemaximumanglecondition.Again,aremarkanalogoustoRemark3.30holds.Forananalogousinvestigationoftetrahedrawereferto[58].Withamodificationoftheaboveconsiderationsandanadditionalconditionanisotropicerrorestimatesoftheformd|v−Ih(v)|1≤Chi|∂iv|1i=1canbeprovenforv∈H2(Ω),wherethehdenotelengthparameterde-ipendingontheelementtype.Inthecaseoftriangles,thesearethelongestedge(h1=hK)andtheheightonitasshowninFigure3.12(see[41]).3.4.3L2ErrorEstimatesTheerrorestimate(3.89)alsocontainsaresultabouttheapproximationofthegradient(andhenceoftheflux),butitislinearonlyfork=1,in 3.4.ConvergenceRateEstimates145contrasttotheerrorestimateofChapter1(Theorem1.6).Thequestioniswhetheranimprovementoftheconvergencerateispossibleifwestriveonlyforanestimateofthefunctionvalues.ThedualityargumentofAubinandNitscheshowsthatthisiscorrect,iftheadjointboundaryvalueproblemisregular,wherewehavethefollowingdefinition:Definition3.36Theadjointboundaryvalueproblemfor(3.12),(3.18)–(3.20)isdefinedbythebilinearform(u,v)→a(v,u)foru,v∈VwithVfrom(3.30).Itiscalledregularifforeveryf∈L2(Ω)thereexistsauniquesolutionu=uf∈Voftheadjointboundaryvalueproblema(v,u)=f,v0forallv∈Vandevenu∈H2(Ω)issatisfied,andforsomeconstantC>0astabilityfestimateoftheform|u|≤Cfforgivenf∈L2(Ω)f20issatisfied.TheV-ellipticityandthecontinuityofthebilinearform(3.2),(3.3)di-rectlycarryoverfrom(3.31)totheadjointboundaryvalueproblem,sothatinthiscasetheuniqueexistenceofuf∈Visensured.Morepre-cisely,theadjointboundaryvalueproblemisobtainedbyanexchangeoftheargumentsinthebilinearform,whichdoesnoteffectanychangeinitssymmetricparts.Thenonsymmetricpartof(3.31)isc·∇uvdx,whichΩbecomesc·∇vudx.ByvirtueofΩ


c·∇vudx=−∇·(cu)vdx+c·νuvdσΩΩ∂Ωthetransitiontotheadjointboundaryvalueproblemthereforemeanstheexchangeoftheconvectivepartc·∇ubyaconvectivepart,nowindiver-genceformandintheoppositedirection−c,namely∇·(−cu),withthecorrepondingmodificationoftheboundarycondition.Hence,ingeneralwemayexpectasimilarregularitybehaviortothatintheoriginalboundaryvalueproblem,whichwasdiscussedinSection3.2.3.Foraregularadjointproblemwegetanimprovementoftheconvergenceratein·0:Theorem3.37(AubinandNitsche)ConsiderthesituationofTheorem3.29orTheorem3.35andsupposetheadjointboundaryvalueproblemisregular.ThenthereexistssomeconstantC>0suchthatforthesolutionuoftheboundaryvalueproblem(3.12),(3.18)–(3.20)anditsfiniteelementapproximationuhdefinedby(3.39),(1)u−uh0≤Chu−uh1,(2)u−uh0≤Chu1, 1463.FiniteElementMethodsforLinearEllipticProblems(3)u−u≤Chk+1|u|,ifu∈Hk+1(Ω).h0k+1Proof:Theassertions(2)and(3)followdirectlyfrom(1).Ontheonehand,byusingu−uh1≤u1+uh1andthestabilityestimate(2.44),ontheotherhanddirectlyfrom(3.89)and(3.98),respectively.Fortheproofof(1),weconsiderthesolutionufoftheadjointproblemwiththeright-handsidef=u−u∈V⊂L2(Ω).Choosingthetesthfunctionu−uhandusingtheerrorequation(2.39)gives2u−uh0=u−uh,u−uh0=a(u−uh,uf)=a(u−uh,uf−vh)forallvh∈Vh.Ifwechoosespecificallyvh=Ih(uf),thenfromthecon-tinuityofthebilinearform,Theorem3.29,andTheorem3.35,andtheregularityassumptionitfollowsthatu−u2≤Cu−uu−I(u)h0h1fhf1≤Cu−uh1h|uf|2≤Cu−uh1hu−uh0.Divisionbyu−uh0givestheassertion,whichistrivialinthecaseu−uh0=0.
Thus,ifaroughright-handsidein(3.12)preventsconvergencefrombeingensuredbyTheorem3.29orTheorem3.35,thentheestimate(2)canstillbeusedtogetaconvergenceestimate(oflowerorder).InthelightoftheconsiderationsfromSection1.2,theresultofTheo-rem3.37issurprising,sincewehaveonly(pointwise)consistencyoffirstorder.Ontheotherhand,Theorem1.6alsoraisesthequestionofconver-gencerateresultsin·∞whichthenwouldgivearesultstronger,inmanyrespects,thanTheorem1.6.Althoughtheconsiderationsdescribedhere(asinSection3.9)canbethestartingpointofsuchL∞estimates,wegetthemostfar-reachingresultswiththeweightednormtechnique(see[9,pp.155ff.]),whosedescriptionisnotpresentedhere.Theabovetheoremscontainconvergencerateresultsunderregularityassumptionsthatmayoften,eventhoughonlylocally,beviolated.Infact,therealsoexist(weaker)resultswithlessregularityassumptions.However,thefollowingobservationseemstobemeaningful:Estimate(3.90)indicatesthatonsubdomains,wherethesolutionhaslessregularity,onwhichthe(semi)normsofthesolutionsthusbecomelarge,localrefinementisadvan-tageous(withoutimprovingtheconvergenceratebythis).AdaptivemeshrefinementstrategiesonthebasisofaposteriorierrorestimatesdescribedinChapter4provideasystematicalapproachinthisdirection.Exercises3.21Proveforthelinearfiniteelementansatz(3.53)inonespacedi-mensionthatforK∈Tandv∈H2(K),thefollowingestimateh Exercises147holds:|v−IK(v)|1,K≤hK|v|2,K.Hint:Rolle’stheoremandExercise2.5(b)(Poincar´einequality).GeneralizetheconsiderationstoanarbitrarypolynomialansatzP=Pkinonespacedimensionbyproving|v−I(v)|≤hk|v|forv∈Hk+1(K).K1,KKk+1,K3.22Provethechainrule(3.77)forv∈H1(K).3.23DeriveanalogouslytoTheorem3.29aconvergencerateresultfortheHermiteelements(3.64)and(3.65)(Bogner–Fox–Schmitelement)andtheboundaryvalueproblem(3.12)withDirichletboundaryconditions.3.24DeriveanalogouslytoTheorem3.29aconvergencerateresultfortheBogner–Fox–Schmitelement(3.65)andtheboundaryvalueproblem(3.36).3.25LetatriangleKwiththeverticesa1,a2,a3andafunctionu∈C2(K)begiven.ShowthatifuisinterpolatedbyalinearpolynomialIK(u)with(IK(u))(ai)=u(ai),i=1,2,3,then,fortheerrortheestimateh2sup|u(x)−(IK(u))(x)|+hsup|∇(u−IK(u))(x)|≤2Mx∈Kx∈Kcos(α/2)holds,wherehdenotesthediameter,αthesizeofthelargestinteriorangleofKandManupperboundforthemaximumofthenormoftheHessianmatrixofuonK.3.26ConsideratriangleKwiththeverticesa1:=(−h,0),a2:=(h,0),a:=(0,ε),andh,ε>0.Supposethatthefunctionu(x):=x2islinearly31interpolatedonKsuchthat(Ih(u))(ai)=u(ai)fori=1,2,3.Determine∂2(Ih(u)−u)2,Kaswellas∂2(Ih(u)−u)∞,Kanddiscusstheconsequencesforofdifferentordersofmagnitudeofhandε.3.27Supposethatnofurtherregularitypropertiesareknownforthesolutionu∈Voftheboundaryvalueproblem(3.12).ShowundertheassumptionsofSection3.4thatforthefiniteelementapproximationuh∈Vhu−uh1→0forh→0. 1483.FiniteElementMethodsforLinearEllipticProblems3.5TheImplementationoftheFiniteElementMethod:Part23.5.1IncorporationofDirichletBoundaryConditions:Part2InthetheoreticalanalysisofboundaryvalueproblemswithinhomogeneousDirichletboundaryconditionsu=g3onΓ3,theexistenceofafunctionw∈H1(Ω)withw=gonΓhasbeenassumedsofar.Thesolution33u∈V(withhomogeneousDirichletboundaryconditions)isthendefinedaccordingto(3.31)suchthat˜u=u+wsatisfiesthevariationalequationwithtestfunctionsinV:a(u+w,v)=b(v)forallv∈V.(3.99)FortheGalerkinapproximationuh,whichhasbeenanalyzedinSection3.4,thismeansthattheparts−a(w,ϕi)withnodalbasisfunctionsϕi,i=1,...,M1,gointotheright-handsideofthesystemofequations(2.34),andthen˜uh:=uh+whastobeconsideredasthesolutionoftheinhomogeneousproblema(uh+w,v)=b(v)forallv∈Vh.(3.100)IfwecompletethebasisofVhbythebasisfunctionsϕM1+1,...,ϕMfortheDirichletboundarynodesaM1+1,...,aManddenotethegeneratedspacebyXh,Xh=span{ϕ1,...,ϕM1,ϕM1+1,...,ϕM},(3.101)thatistheansatzspacewithouttakingintoaccountboundaryconditions,theninparticular,˜uh∈Xhdoesnotholdingeneral.ThisapproachdoesnotcorrespondtothepracticedescribedinSection2.4.3.Thatpractice,appliedtoageneralvariationalequation,readsasfollows:Foralldegreesoffreedom1,...,M1,M1+1,...,Mthesystemofequationsisbuiltwiththecomponentsa(ϕj,ϕi),i,j=1,...,M,(3.102)forthestiffnessmatrixandb(ϕi),i=1,...,M,(3.103)fortheloadvector.Thevectorofunknownsisthereforeξξ˜=withξ∈RM1,ξˆ∈RM2.ξˆForDirichletboundaryconditionstheequationsM1+1,...,Marereplacedbyξ˜i=g3(ai),i=M1+1,...,M, 3.5.TheImplementationoftheFiniteElementMethod:Part2149andtheconcernedvariablesareeliminatedinequations1,...,M1.Ofcourse,itisassumedherethatg3∈C(Γ3).Thisprocedurecanalsobeinterpretedinthefollowingway:IfwesetAh:=(a(ϕj,ϕi))i,j=1,...,M1,Aˆh:=(a(ϕj,ϕi))i=1,...,M1,j=M1+1,...,M,thenthefirstM1equationsofthegeneratedsystemofequationsareAhξ+Aˆhξˆ=qh,whereq∈RM1consistsofthefirstMcomponentsaccordingto(3.103).h1HencetheeliminationleadstoAhξ=qh−Aˆhξˆ(3.104)withξˆ=(g3(ai))i=M1+1,...,M2.SupposeMwh:=g3(ai)ϕi∈Xh(3.105)i=M1+1istheansatzfunctionthatsatisfiestheboundaryconditionsintheDirichletnodesandassumesthevalue0inallothernodes.Thesystemofequations(3.104)isthenequivalenttoa(ˇuh+wh,v)=b(v)forallv∈Vh(3.106)M1forˇuh=i=1ξiϕi∈Vh(thatis,the“real”solution),incontrasttothevariationalequation(3.100)wasusedintheanalysis.Thisconsiderationalsoholdsifanotherh-dependentbilinearformahandanalogouslyalin-earformbhinsteadofthelinearformbisusedforassembling.Inthefollowingweassumethatthereexistssomefunctionw∈C(Ω)thatsat-¯isfiestheboundaryconditiononΓ3.Insteadof(3.106),weconsiderthefinite-dimensionalauxiliaryproblemoffindingsomeuˇˇh∈Vh,suchthata(uˇˇh+I¯h(w),v)=b(v)forallv∈Vh.(3.107)HereI¯h:C(Ω)¯→Xhistheinterpolationoperatorwithrespecttoalldegreesoffreedom,M1+M2I¯h(v):=v(ai)ϕi,i=1whereasinSection3.4weconsideredtheinterpolationoperatorIhforfunc-tionsthatvanishonΓ3.Inthefollowing,whenanalyzingtheeffectofquadrature,wewillshowthat—alsoforsomeapproximationofaandb—u˜h:=uˇˇh+I¯h(w)∈Xh(3.108)isanapproximationofu+wofthequalityestablishedinTheorem3.29(seeTheorem3.42).Wehavewh−I¯h(w)∈Vhandhencealsoˇuh+wh− 1503.FiniteElementMethodsforLinearEllipticProblemsI¯h(w)∈Vh.If(3.107)isuniquelysolvable,whichfollowsfromthegeneralassumptionoftheV-ellipticityofa(3.3),wehaveuˇh+wh−I¯h(w)=uˇˇhandhencefor˜uh,accordingto(3.108),u˜h=ˇuh+wh.(3.109)InthiswaythedescribedimplementationpracticeforDirichletboundaryconditionsisjustified.3.5.2NumericalQuadratureWeconsideragainaboundaryvalueprobleminthevariationalformulation(3.31)andafiniteelementdiscretizationinthegeneralformdescribedinSections3.3and3.4.IfwestepthroughSection2.4.2describingtheassemblingwithinafiniteelementcode,wenoticethatthegeneralelement-to-elementapproachwithtransformationtothereferenceelementisherealsopossible,withtheexceptionthatduetothegeneralcoefficientfunctionsK,c,randf,thearisingintegralscannotbeevaluatedexactlyingeneral.IfKmisageneralelementwithdegreesoffreedominar1,...,arL,thenthecomponentsoftheelementstiffnessmatrixfori,j=1,...,Lare
(m)Aij=K∇ϕrj·∇ϕri+c·∇ϕrjϕri+rϕrjϕridxKm
+αϕrjϕridσ(3.110)Km∩Γ2

=:vij(x)dx+wij(σ)dσKmKm∩Γ2

=vˆij(ˆx)dxˆ|det(B)|+wˆij(ˆσ)dσˆ|det(B˜)|.KˆKˆHere,KmisaffineequivalenttothereferenceelementKˆbythemappingF(ˆx)=Bxˆ+d.Byvirtueoftheconformityofthetriangulation(T6),theboundarypartKm∩Γ¯2consistsofnone,one,ormorecompletefacesofKm.Forsimplicity,werestrictourselvestothecaseofonefacethatisaffineequivalenttothereferenceelementKˆ
bysomemappingF˜(ˆσ)=B˜σˆ+d˜(cf.(3.42)).Thegeneralizationtotheothercasesisobvious.Thefunctionsvˆijandanalogouslyˆwijarethetransformedfunctionsdefinedin(3.75).Correspondingly,wegetascomponentsfortheright-handsideofthesystemofequations,thatis,fortheloadvector,
q(m)=fˆ(ˆx)N(ˆx)dxˆ|det(B)|(3.111)iiKˆ

+gˆ1(ˆσ)Ni(ˆσ)dσˆ|det(B˜1)|+gˆ2(ˆσ)Ni(ˆσ)dσˆ|det(B˜2)|.KˆKˆ12 3.5.TheImplementationoftheFiniteElementMethod:Part2151i=1,...,L.Here,theNi,i=1,...,L,aretheshapefunctions;thatis,thelocalnodalbasisfunctionsonKˆ.Ifthetransformedintegrandscontainderivativeswithrespecttox,theycanbetransformedintoderivativeswithrespecttoˆx.Forinstance,forthe(m)firstaddendinAijweget,asanextensionof(2.50),
K(F(ˆx))B−T∇N(ˆx)·B−T∇N(ˆx)dxˆ|det(B)|.xˆjxˆiKˆTheshapefunctions,theirderivatives,andtheirintegralsoverKˆareknownwhichhasbeenusedin(2.52)fortheexactintegration.Sincegeneralcoef-ficientfunctionsarise,thisisingeneral,butalsointheremainingspecialcasesnolongerpossible,forexampleforpolynomialK(x)itisalsonotrecommendableduetothecorrespondingeffort.Instead,oneshouldap-proximatetheseintegrals(and,analogously,alsotheboundaryintegrals)byusingsomequadratureformula.AquadratureformulaonKˆfortheapproximationofKˆvˆ(ˆx)dxˆhastheformRωˆivˆ(ˆbi)(3.112)i=1withweightsωˆiandquadratureorintegrationpointsˆbi∈K.ˆHence,ap-plying(3.112)assumestheevaluabilityofˆvinˆbi,whichisinthefollowingensuredbythecontinuityofˆv.Thisimpliesthesameassumptionforthecoefficients,sincetheshapefunctionsNiandtheirderivativesarecontinu-ous.Inordertoensurethenumericalstabilityofaquadratureformula,itisusuallyrequiredthatωˆi>0foralli=1,...,R,(3.113)whichwewillalsodo.Sincealltheconsideredfiniteelementsaresuchthattheirfaceswiththeencloseddegreesoffreedomrepresentagainafi-niteelement(inRd−1)(see(3.42)),theboundaryintegralsareincludedinageneraldiscussion.Inprinciple,differentquadratureformulascanbeappliedforeachoftheaboveintegrals,butherewewilldisregardthispos-sibility(withtheexceptionofdistinguishingbetweenvolumeandboundaryintegralsbecauseoftheirdifferentdimensions).AquadratureformulaonKˆgeneratesaquadratureformulaonageneralelementK,recalling

v(x)dx=vˆ(ˆx)dxˆ|det(B)|KKˆbyRωi,Kv(bi,K),i=1 1523.FiniteElementMethodsforLinearEllipticProblemswhereωi=ωi,K=ˆωi|det(B)|andbi=bi,K:=F(ˆbi)aredependentonK.Thepositivityoftheweightsispreserved.Here,againF(ˆx)=Bxˆ+ddenotestheaffine-lineartransformationfromKˆtoK.Theerrorsofthequadratureformulas
REˆ(ˆv):=vˆ(ˆx)dxˆ−ωˆivˆ(ˆbi),Kˆi=1(3.114)
REK(v):=v(x)dx−ωiv(bi)Ki=1arerelatedtoeachotherbyEK(v)=|det(B)|Eˆ(ˆv).(3.115)Theaccuracyofaquadratureformulawillbedefinedbytherequirementthatforlaslargeaspossible,Eˆ(ˆp)=0forˆp∈Pl(Kˆ)issatisfied,whichtransfersdirectlytotheintegrationoverK.Aquadratureformulashouldfurtherprovidethedesiredaccuracybyusingquadraturenodesaslessaspossible,sincetheevaluationofthecoefficientfunctionsisoftenexpensive.Incontrast,fortheshapefunctionsandtheirderivativesasingleevaluationissufficient.Inthefollowingwediscusssomeexam-plesofquadratureformulasfortheelementsthathavebeenintroducedinSection3.3.Themostobviousapproachconsistsinusingnodalquadratureformu-las,whichhavethenodesˆa1,...,aˆLofthereferenceelement(K,ˆP,ˆΣ)asˆquadraturenodes.TherequirementofexactnessinPˆisthenequivalentto
ωˆi=Ni(ˆx)dx,ˆ(3.116)Kˆsothatthequestionofthevalidityof(3.113)remains.WestartwiththeunitsimplexKˆdefinedin(3.47).Here,theweightsofthequadratureformulascanbegivendirectlyonageneralsimplexK:Iftheshapefunctionsareexpressedbytheirbarycentriccoordinatesλi,theintegralscanbecomputedby
α1α2αd+1α1!α2!···αd+1!vol(K)λ1λ2···λd+1(x)dx=(3.117)K(α1+α2+···+αd+1+d)!vol(Kˆ)(seeExercise3.28).IfP=P1(K)andthusthequadraturenodesarethevertices,itfollowsthat
1ωi=λi(x)dx=vol(K)foralli=1,...,d+1.(3.118)Kd+1 3.5.TheImplementationoftheFiniteElementMethod:Part2153ForP=P2(K)andd=2weget,bytheshapefunctionsλi(2λi−1),theweights0forthenodesaiand,bytheshapefunctions4λiλj,theweights1ωi=vol(K)forbi=aij,i,j=1,...,3,i>j,3sothatwehaveobtainedhereaquadratureformulathatissuperiorto(3.118)(ford=2).However,ford≥3thisansatzleadstonegativeweightsandisthususeless.WecanalsogettheexactnessinP1(K)byasinglequadraturenode,bythebarycentre(see(3.52)):d+11ω1=vol(K)andb1=aS=ai,d+1i=1whichisobviousdueto(3.117).AsaformulathatisexactforP2(K)andd=3(see[53])wepresentR=4,ω=1vol(K),andthebareobtainedbycyclicexchangeofthei4ibarycentriccoordinates:√√√√5−55−55−55+35,,,.20202020OntheunitcuboidKˆweobtainnodalquadratureformulas,whichareexactforQk(Kˆ),fromtheNewton–Cˆotesformulasintheone-dimensionalsituationbyi1idωˆi1...id=ˆωi1···ωˆidforˆbi1...id=,...,(3.119)kkforij∈{0,...,k}andj=1,...,d.1HeretheˆωijaretheweightsoftheNewton–Cˆotesformulafor0f(x)dx(see[30,p.128]).Asin(3.118),fork=1wehavehereageneralizationofthetrapezoidalrule(cf.(2.38),(8.31))withtheweights2−dinthe2dvertices.Fromk=8on,negativeweightsarise.ThiscanbeavoidedandtheaccuracyforagivennumberofpointsincreasediftheNewton–CˆotesintegrationisreplacedbytheGauss–(Legendre)integration:In(3.119),ij/khastobereplacedbythejthnodeofthekthGauss–Legendreformula(see[30,p.156]thereon[−1,1])andanalogouslyˆωij.Inthisway,by(k+1)dquadraturenodestheexactnessinQ(Kˆ),notonlyinQ(Kˆ),2k+1kisobtained.Nowthequestionastowhichquadratureformulashouldbechosenarises.Forthis,differentcriteriacanbeconsidered(seealso(8.29)).Here,were-quirethattheconvergencerateresultthatwasprovedinTheorem3.29shouldnotbedeteriorated.Inordertoinvestigatethisquestionwehavetoclarifywhichproblemissolvedbytheapproximation¯uh∈Vhbasedonquadrature.Tosimplifythenotation,fromnowonwedonotconsiderboundaryintegrals,thatis,onlyDirichletandhomogeneousNeumann 1543.FiniteElementMethodsforLinearEllipticProblemsboundaryconditionsareallowed.However,thegeneralizationshouldbeclear.Replacingtheintegralsin(3.111)and(3.111)byquadratureformu-Rlasi=1ωˆivˆ(ˆbi)leadstosomeapproximationA¯hofthestiffnessmatrixandq¯hoftheloadvectorintheformA¯h=(ah(ϕj,ϕi))i,j,q¯h=(bh(ϕi))i,fori,j=1,...,M.HeretheϕiarethebasisfunctionsofXh(see(3.101))withouttakingintoaccounttheDirichletboundaryconditionandRah(v,w):=ωl,K(K∇v·∇w)(bl,K)K∈Thl=1R+ωl,K(c·∇vw)(bl,K)+ωl,K(rvw)(bl,K)K∈Thl=1K∈Thl=1forv,w∈Xh,(3.120)bh(v):=ωl,K(fv)(bl,K)forv∈Xh.K∈Thl=1Theabove-givenmappingsahandbharewell-definedonXh×XhandXh,respectively,ifthecoefficientfunctionscanbeevaluatedinthequadraturenodes.HerewetakeintoaccountthatforsomeelementK,∇vforv∈Xhcanhavejumpdiscontinuitieson∂K.Thus,forthequadraturenodesbl,K∈∂Kin∇v(bl,K)wehavetochoosethevalue“belongingtobl,K”thatcorrespondstothelimitofsequencesintheinteriorofK.WerecallthatingeneralahandbharenotdefinedforfunctionsofV.Obviously,ahisbilinearandbhislinear.IfwetakeintoaccounttheanalysisofincorporatingtheDirichletboundaryconditionsin(3.99)–(3.106),wegetasystemofequationsforthedegreesoffreedomξ¯=(ξ,...,ξ)T,whichisequivalent1M1M1ξ¯tothevariationalequationonVhfor¯uh=i=1iϕi∈Vh:ah(¯uh,v)=bh(v)−ah(wh,v)forallv∈Vh(3.121)withwhaccordingto(3.105).Ashasbeenshownin(3.109),(3.121)isequivalent,inthesenseofthetotalapproximation¯uh+whofu+w,tothevariationalequationforu¯¯h∈Vh,ah(u¯¯h,v)=¯bh(v):=bh(v)−ah(I¯h(w),v)forallv∈Vh,(3.122)ifthissystemofequationsisuniquelysolvable.Exercises3.28Proveequation(3.117)byfirstprovingtheequationforK=KˆandthendeducingfromthistheassertionforthegeneralsimplexbyExercise3.18. 3.6.ConvergenceRateResultsinCaseofQuadratureandInterpolation1553.29LetKbeatrianglewithverticesa1,a2,a3.Further,leta12,a13,a23denotethecorrespondingedgemidpoints,a123thebarycenterand|K|theareaofK.Checkthatthequadratureformula01|K|3Qh(u):=3u(ai)+8u(aij)+27u(a123)60i=1i0,uh∈Vhsatisfiesah(uh,v)=lh(v)forallv∈Vh.(3.124)HereaandaharebilinearformsonV×VandVh×Vh,respectively,andl,lharelinearformsonVandVh,respectively.ThenwehavethefollowingtheoremTheorem3.38(FirstLemmaofStrang)Supposethereexistssomeα>0suchthatforallh>0andv∈Vh,αv2≤a(v,v),(3.125)handletabecontinuousinV×V.Then,thereexistssomeconstantCindependentofVhsuchthat|a(v,w)−ah(v,w)|u−uh≤Cinfu−v+supv∈Vhw∈Vhw|l(w)−lh(w)|+sup.w∈Vhw(3.126) 1563.FiniteElementMethodsforLinearEllipticProblemsProof:Letv∈Vhbearbitrary.Thenitfollowsfrom(3.123)–(3.125)thatαu−v2≤a(u−v,u−v)hhhh=a(u−v,uh−v)+a(v,uh−v)−ah(v,uh−v)+lh(uh−v)−l(uh−v)andmoreover,bythecontinuityofa(cf.(3.2)),|a(v,w)−ah(v,w)|αuh−v≤Mu−v+supw∈Vhw|lh(w)−l(w)|+supforv∈Vh.w∈VhwBymeansofu−uh≤u−v+uh−vandtakingtheinfimumoverallv∈Vh,theassertionfollows.
Forah=aandlh=ltheassertionreducestoC´ea’slemma(Theo-rem2.17),whichwastheinitialpointfortheanalysisoftheconvergencerateinSection3.4.Herewecanproceedanalogously.Forthatpurpose,thefollowingconditionsmustbefulfilledadditionally:•TheuniformVh-ellipticityofahaccordingto(3.125)mustbeensured.•Fortheconsistencyerrors|a(v,w)−ah(v,w)|Ah(v):=sup(3.127)w∈Vhwforanarbitrarilychosencomparisonfunctionv∈Vhandfor|l(w)−lh(w)|supw∈Vhwthebehaviorinhmustbeanalyzed.ThefirstrequirementisnotcrucialifonlyaitselfisV-ellipticandAhtendssuitablyto0forh→0:Lemma3.39SupposethebilinearformaisV-ellipticandthereexistssomefunctionC(h)withC(h)→0forh→0suchthatAh(v)≤C(h)vforv∈Vh.Thenthereexistssomeh>¯0suchthatahisuniformlyVh-ellipticforh≤h¯.Proof:Byassumption,thereexistssomeα>0suchthatforv∈Vh,αv2≤a(v,v)+a(v,v)−a(v,v)hhand2|a(v,v)−ah(v,v)|≤Ah(v)v≤C(h)v. 3.6.ConvergenceRateResultsinCaseofQuadratureandInterpolation157Therefore,forinstance,chooseh¯suchthatC(h)≤α/2forh≤h¯.
Weconcretelyaddresstheanalysisoftheinfluenceofnumericalquadra-ture,thatis,ahisdefinedasin(3.120)andlhcorrespondsto¯bhin(3.122)withtheapproximatelinearformbhaccordingto(3.120).Sincethisisanextensionoftheconvergenceresults(in·1)giveninSection3.4,theas-sumptionsaboutthefiniteelementdiscretizationareassummarizedthereatthebeginning.Inparticular,thetriangulationsThconsistofelementsthatareaffineequivalenttoeachother.Furthermore,forasimplificationofthenotation,letagaind≤3andonlyLagrangeelementsareconsidered.Inparticular,letthegeneralassumptionsabouttheboundaryvalueproblemswhicharespecifiedattheendofSection3.2.1besatisfied.AccordingtoTheorem3.38,theuniformVh-ellipticityofahmustbeensuredandtheconsistencyerrors(foranappropriatecomparisonelementv∈Vh)musthavethecorrectconvergencebehavior.Ifthestepsizehissmallenough,thefirstpropositionisimpliedbythesecondpropositionbyvirtueofLemma3.39.Now,simplecriteriathatareindependentofthisrestrictionwillbepresented.Thequadratureformulassatisfytheproperties(3.112),(3.113)introducedinSection3.5;inparticular,theweightsarepositive.Lemma3.40SupposethecoefficientfunctionKsatisfies(3.16)andletc=0inΩ,let|Γ3|d−1>0,andletr≥0inΩ.IfP⊂Pk(K)fortheansatzspaceandifthequadratureformulaisexactforP2k−2(K),thenahisuniformlyVh-elliptic.Proof:Letα>0betheconstantoftheuniformpositivedefinitenessofK(x).Thenwehaveforv∈Vh:R
a(v,v)≥αω|∇v|2(b)=α|∇v|2(x)dx=α|v|2,hl,Kl,K1ΩK∈Thl=1since|∇v|2∈P(K).TheassertionfollowsfromCorollary3.14.
K2k−2Furtherresultsofthistypecanbefoundin[9,pp.194].ToinvestigatetheconsistencyerrorwecanproceedsimilarlytotheestimationoftheinterpolationerrorinSection3.4:TheerrorissplitintothesumoftheerrorsovertheelementsK∈Thandtheretransformedbymeansof(3.115)intotheerroroverthereferenceelementKˆ.Thederivatives(inˆx)arisingintheerrorestimationoverKˆarebacktransformedbyusingTheorem3.26andTheorem3.27,whichleadstothedesiredhK-factors.ButnotethatpowersofB−1orsimilartermsdonotarise.Ifthepowersofdet(B)arisinginbothtransformationstepscanceleachother(whichwillhappen),inthiswaynoconditionaboutthegeometricqualityofthefamilyoftriangulationsarises.Ofcourse,theseresultsmustbecombinedwithestimatesforthe 1583.FiniteElementMethodsforLinearEllipticProblemsapproximationerrorofVh,forwhich,inparticular,bothapproachesofSection3.4(eitherregularityormaximumanglecondition)areadmissible.Forthesakeofsimplicity,werestrictourattentioninthefollowingtothecaseofthepolynomialansatzspaceP=Pk(K).Moregeneralresultsofsimilartype,inparticularfortriangulationswiththecuboidelementandPˆ=Qk(Kˆ)asreferenceelement,aresummarizedin[9,p.207].Werecallthenotationandtherelationsintroducedin(3.114),(3.115)forthelocalerrors.InthefollowingtheoremswemakeuseoftheSobolevspacesWlonΩandonKwiththenorms·and·,respectively,∞l,∞l,∞,Kandtheseminorms|·|l,∞and|·|l,∞,K,respectively.Theessentiallocalassertionisthefollowing:Theorem3.41Supposek∈NandPˆ=Pk(Kˆ)andthequadratureformulaisexactforP2k−2(Kˆ):Eˆ(ˆv)=0forallvˆ∈P2k−2(Kˆ).(3.128)ThenthereexistsomeconstantC>0independentofh>0andK∈Thsuchthatforl∈{1,k}thefollowingestimatesaregiven:(1)|E(apq)|≤ChlapqKKk,∞,Kl−1,K0,Kfora∈Wk(K),p,q∈P(K),∞k−1(2)|E(cpq)|≤ChlcpqKKk,∞,Kl−1,K1,Kkforc∈W∞(K),p∈Pk−1(K),q∈Pk(K),(3)|E(rpq)|≤ChlrpqKKk,∞,Kl,K1,Kkforr∈W∞(K),p,q∈Pk(K),(4)|E(fq)|≤Chkfvol(K)1/2qKKk,∞,K1,Kkforf∈W∞(K),q∈Pk(K).The(unnecessarilyvaried)notationofthecoefficientsalreadyindicatesthefieldofapplicationoftherespectiveestimate.Thesmoothnessassump-tionconcerningthecoefficientsin(1)–(3)canbeweakenedtosomeextent.Weproveonlyassertion(1).However,adirectapplicationofthisprooftoassertions(2)–(4)leadstoalossofconvergencerate(orhigherexactnessconditionsforthequadrature).Here,quitetechnicalconsiderationsinclud-ingtheinsertionofprojectionsarenecessary,whichcanbefoundtosomeextentin[9,pp.201–203].Inthefollowingproofweintensivelymakeuseofthefactthatallnormsareequivalentonthe“fixed”finite-dimensionalansatzspacePk(Kˆ).Theassumption(3.128)isequivalenttothesamecon-ditiononageneralelement.However,theformulationalreadyindicatesanassumptionthatisalsosufficientinmoregeneralcases. 3.6.ConvergenceRateResultsinCaseofQuadratureandInterpolation159ProofofTheorem3.41,(1):WeconsiderageneralelementK∈Thandmappingsa∈Wk(K),p,q∈P(K)onitand,moreover,mappings∞k−1aˆ∈Wk(Kˆ),ˆp,qˆ∈P(Kˆ)definedaccordingto(3.75).First,theproof∞k−1isdoneforl=k.OnthereferenceelementKˆ,forˆv∈Wk(Kˆ)andˆq∈∞Pk−1(Kˆ)),wehave
REˆ(ˆvqˆ)=vˆqdˆxˆ−ωˆ(ˆvqˆ)(ˆb)≤Cvˆqˆ≤Cvˆqˆ,ll∞,Kˆ∞,Kˆ∞,KˆKˆl=1wherethecontinuityoftheembeddingofWk(Kˆ)inC(Kˆ)isused(see[8,∞p.181]).Therefore,bytheequivalenceof·∞,Kˆand·0,KˆonPk−1(Kˆ),itfollowsthatEˆ(ˆvqˆ)≤Cvˆqˆ.k,∞,Kˆ0,KˆIfafixedˆq∈Pk−1(Kˆ)ischosen,thenalinearcontinuousfunctionalGisdefinedonWk(Kˆ)byˆv→Eˆ(ˆvqˆ)thathasthefollowingproperties:∞G≤Cqˆ0,KˆandG(ˆv)=0forˆv∈Pk−1(Kˆ)byvirtueof(3.128).TheBramble–Hilbertlemma(Theorem3.24)impliesEˆ(ˆvqˆ)≤C|vˆ|qˆ.k,∞,Kˆ0,KˆAccordingtotheassertionwenowchoosevˆ=ˆapˆforaˆ∈Wk,∞(Kˆ),pˆ∈P(Kˆ),k−1andwehavetoestimate|aˆpˆ|(thankstotheBramble–Hilbertlemmak,∞Kˆnotaˆpˆ).TheLeibnizruleforthedifferentiationofproductsimpliesk,∞,Kˆtheestimatek|aˆpˆ|≤C|aˆ||pˆ|.(3.129)k,∞,Kˆk−j,∞,Kˆj,∞,Kˆj=0HeretheconstantCdependsonlyonk,butnotonthedomainKˆ.Sinceˆp∈Pk−1(Kˆ),thelasttermofthesumin(3.129)canbeomitted.Therefore,wehaveobtainedthefollowingestimateholdingforˆa∈Wk(Kˆ),∞p,ˆqˆ∈Pk−1(Kˆ):$%k−1Eˆ(ˆapˆqˆ)≤C|aˆ||pˆ|qˆk−j,∞,Kˆj,∞,Kˆ0,Kˆj=0$%(3.130)k−1≤C|aˆ||pˆ|qˆ.k−j,∞,Kˆj,Kˆ0,Kˆj=0Thelastestimateusestheequivalenceof·∞and·0onPk−1(Kˆ).WesupposethatthetransformationFofKˆtothegeneralelementKhas,asusual,thelinearpartB.Thefirsttransformationstepyieldsthe 1603.FiniteElementMethodsforLinearEllipticProblemsfactor|det(B)|accordingto(3.115),andforthebacktransformationitfollowsfromTheorem3.26andTheorem3.27thatk−j|aˆ|k−j,∞,Kˆ≤ChK|a|k−j,∞,K,|pˆ|≤Chj|det(B)|−1/2|p|,(3.131)j,KˆKj,Kqˆ≤C|det(B)|−1/2q0,Kˆ0,Kfor0≤j≤k−1.Herea,p,qarethemappingsˆa,p,ˆqˆ(back)transformedaccordingto(3.75).Substitutingtheseestimatesinto(3.130)thereforeyields$%k−1E(apq)≤Chk|a||p|qKKk−j,∞,Kj,K0,Kj=0andfromthis,assertion(1)followsforl=k.Ifl=1,wemodifytheproofasfollows.Again,in(3.130)weestimatebyusingtheequivalenceofnorms:$%k−1Eˆ(ˆapˆqˆ)≤C|aˆ|pˆqˆk−j,∞,Kˆj,∞,Kˆ0,Kˆj=0$%k−1≤C|aˆ|pˆqˆ.k−j,∞,Kˆ0,Kˆ0,Kˆj=0Thefirstandthethirdestimatesof(3.131)remainapplicable;thesecondestimateisreplacedwiththethirdsuchthatwehave$%k−1EK(apq)≤ChK|a|k−j,∞,K,p0,Kq0,Kj=0sincethelowesthK-powerarisesforj=k−1.Thisestimateyieldstheassertion(1)forl=1.
Finally,wecannowverifytheassumptionsofTheorem3.38withthefollowingresult:Theorem3.42ConsiderafamilyofaffineequivalentLagrangefiniteel-ementdiscretizationsinRd,d≤3,withP=Pforsomek∈Naslocalkansatzspace.Supposethatthefamilyoftriangulationsisregularorsat-isfiesthemaximumangleconditioninthecaseoftriangleswithk=1.SupposethattheappliedquadratureformulasareexactforP2k−2.LetthefunctionwsatisfyingtheDirichletboundaryconditionandletthesolutionuoftheboundaryvalueproblem(3.12),(3.18)–(3.20)(withg3=0)belongtoHk+1(Ω).ThenthereexistsomeconstantsC>0,h>¯0independentofuandwsuchthatforthefiniteelementapproximationu¯h+whaccordingto(3.105), 3.6.ConvergenceRateResultsinCaseofQuadratureandInterpolation161(3.121),itfollowsforh≤h¯that$du+w−(¯u+w)≤Chk|u|+|w|+khh1k+1k+1ijk,∞i,j=1%d+cik,∞+rk,∞uk+1+wk+1+fk,∞.i=1Proof:Accordingto(3.108),weaimatestimatingu+w−(u¯¯h+I¯h(w)),1whereu¯¯hsatisfies(3.122).ByvirtueofTheorem3.29orTheorem3.35(setformallyΓ3=∅)wehavew−I¯(w)≤Chk|w|.(3.132)h1k+1Forthebilinearformahdefinedin(3.120),itfollowsfromTheorem3.41forv,w∈Vhandl∈{0,k}that$da(v,w)−ah(v,w)≤EK(kij∂j(v|K)∂i(w|K))(3.133)K∈Thi,j=1%d+EK(ci∂i(v|K)w)+EK(rvw)i=1$%dd≤Chlk+c+rKijk,∞,Kik,∞,Kk,∞,KK∈Thi,j=1i=1×vl,Kw1,K$%dd≤Chlk+c+rijk,∞ik,∞k,∞i,j=1i=11/2×v2w,l,K1K∈Thbyestimatingthe·k,∞,K-normsintermsofnormsonthedomainΩandthenapplyingtheCauchy–Schwarzinequalitywith“index”K∈Th.Fromthisweobtainforl=1anestimateoftheform|a(v,w)−ah(v,w)|≤Chv1w1suchthattheestimaterequiredinLemma3.39holds(withC(h)=C·h).Therefore,thereexistssome¯h>0suchthatahisuniformlyVh-ellipticforh≤h¯.Hence,theestimate(3.126)isapplicable,andthefirstaddend,theapproximationerror,behavesasassertedaccordingtoTheorem3.29orTheorem3.35(again,choosev=Ih(u)forthecomparisonelement).Inordertoestimatetheconsistencyerrorofah,acomparisonelementv∈Vhhastobefoundforwhichthecorrespondingpartofthenormin 1623.FiniteElementMethodsforLinearEllipticProblems(3.133)isuniformlybounded.Thisissatisfiedforv=Ih(u),since1/2$%1/2I(u)2≤u+u−I(u)2hk,Kkhk,KK∈ThK∈Th≤uk+Ch|u|k+1≤uk+1duetoTheorem3.29orTheorem3.35.Hence,theconsistencyerrorinabehavesasassertedaccordingto(3.133),sothatonlytheconsistencyerroroflhastobeinvestigated:Wehavel−lh=b−bh−a(w,·)+ah(I¯h(w),·),wherebhisdefinedin(3.120).Ifv∈Vh,thena(w,v)−ah(I¯h(w),v)≤a(w,v)−a(I¯h(w),v)+a(I¯h(w),v)−ah(I¯h(w),v).Forthefirstaddendthecontinuityofaimpliesa(w,v)−a(I¯h(w),v)≤Cw−I¯h(w)1v1,sothatthecorrespondingconsistencyerrorpartbehaveslikew−I¯h(w),1whichhasalreadybeenestimatedin(3.132).Thesecondaddendjustcor-respondstotheestimateusedfortheconsistencyerrorina(here,thedifferencebetweenIhandI¯hisirrelevant),sothatthesamecontributiontotheconvergencerate,nowwithuk+1replacedbywk+1,arises.Finally,Theorem3.41,(4)yieldsforv∈Vh,|b(v)−b(v)|≤|E(fv)|≤Chkvol(K)1/2fvhKKk,∞,K1,KK∈ThK∈Th≤Chk|Ω|1/2fvk,∞1byproceedingasin(3.133).Thisimpliesthelastpartoftheassertedestimate.
IftheuniformVh-ellipticityofahisensuredinadifferentway(perhapsbyLemma3.40),onecandispensewiththesmallnessassumptionabouth.IfestimatesasgiveninTheorem3.41arealsoavailableforothertypesofelements,thentriangulationsconsistingofcombinationsofvariouselementscanalsobeconsidered. 3.7.TheConditionNumberofFiniteElementMatrices1633.7TheConditionNumberofFiniteElementMatricesThestabilityofsolutionalgorithmsforlinearsystemsofequationsasde-scribedinSection2.5dependsontheconditionnumberofthesystemmatrix(see[28,Chapter1]).Theconditionnumberalsoplaysanimpor-tantrolefortheconvergencebehaviorofiterativemethods,whichwillbediscussedinChapter5.Therefore,inthissectionweshallestimatethespectralconditionnumber(seeAppendixA.3)ofthestiffnessmatrixA=a(ϕj,ϕi)(3.134)i,j=1,...,Mandalsoofthemassmatrix(see(7.45))B=ϕj,ϕi0i,j=1,...,M,(3.135)whichisofimportancefortime-dependentproblems.Again,weconsiderafiniteelementdiscretizationinthegeneralformofSection3.4restrictedtoLagrangeelements.Inordertosimplifythenotation,weassumetheaffineequivalenceofallelements.Furtherwesupposethat•thefamily(Th)hoftriangulationsisregular.WeassumethatthevariationalformulationoftheboundaryvalueproblemleadstoabilinearformathatisV-ellipticandcontinuousonV⊂H1(Ω).Asamodificationofdefinition(1.18),letthefollowingnorm(whichisalsoinducedbyascalarproduct)bedefinedintheansatzspaceVh=span{ϕ1,...,ϕM}:1/2d2v0,h:=hK|v(ai)|.(3.136)K∈Thai∈KHere,a1,...,aMdenotethenodesofthedegreesoffreedom,whereinordertosimplifythenotation,MinsteadofM1isusedforthenumberofdegreesoffreedom.Thenormpropertiesfollowdirectlyfromthecorrespondingpropertiesof|·|2exceptforthedefiniteness.ButthedefinitenessfollowsfromtheuniquenessoftheinterpolationprobleminVhwithrespecttodegreesoffreedomai.Theorem3.43(1)ThereexistconstantsC1,C2>0independentofhsuchthatforv∈Vh:C1v0≤v0,h≤C2v0.(2)ThereexistsaconstantC>0independentofhsuchthatforv∈Vh,−1v1≤CminhKv0.K∈Th 1643.FiniteElementMethodsforLinearEllipticProblemsProof:AsalreadyknownfromSections3.4and3.6,theproofisdonelocallyinK∈ThandtheretransformedtothereferenceelementKˆbymeansofF(ˆx)=Bxˆ+d.Ad(1):AllnormsareequivalentonthelocalansatzspacePˆ,thusalso·andtheEuclideannorminthedegreesoffreedom.Hence,there0,KˆexistsomeCˆ1,Cˆ2>0suchthatforˆv∈Pˆ,1/2LCˆvˆ≤|vˆ(ˆa)|2≤Cˆvˆ.10,Kˆi20,Kˆi=1Here,ˆa1,...,aˆLarethedegreesoffreedominKˆ.Dueto(3.50)wehavevol(K)=vol(Kˆ)|det(B)|,andaccordingtothedefinitionofhKandtheregularityofthefamily(Th)h,thereexistconstantsC˜i>0independentofhsuchthatC˜hd≤C˜d≤|det(B)|≤C˜hd.1K3K2KBythetransformationrulewethusobtainforv∈PK,theansatzspaceonK,that1/2LCˆv=Cˆ|det(B)|1/2vˆ≤C˜hd1/2|vˆ(ˆa)|210,K10,Kˆ2Kii=11/2L1/21/2=C˜1/2hd|v(a)|2=C˜hd|vˆ(ˆa)|22Ki2Kiai∈Ki=11/21/2≤C˜hdCˆvˆ=C˜hdCˆ|det(B)|−1/2v2K20,Kˆ2K20,K≤C˜1/2CˆC˜−1/2v.2210,KThisimpliesassertion(1).Ad(2):Arguingasbefore,nowusingtheequivalenceof·and1,Kˆ·0,KˆinPˆ,itfollowsbyvirtueof(3.86)forv∈PK(withthegenericconstantC)thatv≤C|det(B)|1/2B−1vˆ≤CB−1v≤Ch−1v1,K20,Kˆ20,KK0,KbyTheorem3.27andtheregularityof(Th)h,andfromthis,theassertion(2).
Inordertomakethenorm·0,hcomparablewiththe(weighted)Euclideannormweassumeinthefollowing:•ThereexistsaconstantCA>0independentofhsuchthatforeverynodeofTh,thenumberofelements(3.137)towhichthisnodebelongsisboundedbyCA. 3.7.ConditionNumberofFiniteElementMatrices165Thisconditionis(partly)redundant:Ford=2andtriangularelements,theconditionfollowsfromtheuniformlowerbound(3.93)forthesmallestangleasanimplicationoftheregularity.Notethattheconditionneednotbesatisfiedifonlythemaximumangleconditionisrequired.Ingeneral,ifC∈RM,Misamatrixwithrealeigenvaluesλ≤···≤1λMandanorthonormalbasisofeigenvectorsξ1,...,ξM,forinstanceasymmetricmatrix,thenitfollowsforξ∈RM{0}thatTξCξλ1≤T≤λM,(3.138)ξξandtheboundsareassumedforξ=ξ1andξ=ξM.Theorem3.44ThereexistsaconstantC>0independentofhsuchthatwehavedhκ(B)≤CminhKK∈ThforthespectralconditionnumberofthemassmatrixBaccordingto(3.135).Proof:κ(B)=λ/λmustbedetermined.Forarbitraryξ∈RM{0}M1wehaveTTv2ξBξξBξ0,h=,ξTξv2ξTξ0,hMTwherev:=i=1ξiϕi∈Vh.ByvirtueofξBξ=v,v0,thefirstfactorontheright-handsideisuniformlyboundedfromaboveandbelowaccordingtoTheorem3.43.Further,by(3.137)andξ=(v(a),...,v(a))Titfollows1Mthatminhd|ξ|2≤v2≤Chd|ξ|2,K0,hAK∈Thand,thusthesecondfactorisestimatedfromaboveandbelow.Thisleadstoestimatesofthetypeλ≥Cminhd,λ≤Chd,11KM2K∈Thandfromthis,theassertionfollows.
Therefore,ifthefamilyoftriangulations(Th)hisquasi-uniforminthesensethatthereexistsaconstantC>0independentofhsuchthath≤ChKforallK∈Th,(3.139)thenκ(B)isuniformlybounded.Inordertobeabletoargueanalogouslyforthestiffnessmatrix,weassumethatwestayclosetothesymmetriccase: 1663.FiniteElementMethodsforLinearEllipticProblemsTheorem3.45SupposethestiffnessmatrixA(3.134)admitsrealeigen-valuesandabasisofeigenvectors.ThenthereexistsaconstantC>0independentofhsuchthatthefollowingestimatesforthespectralconditionnumberκhold:−2κ(B−1A)≤Cminh,KK∈Th−2κ(A)≤CminhKκ(B).K∈ThProof:Withthenotationof(3.138),weproceedanalogouslytotheproofofTheorem3.44.SinceTTTξAξξAξξBξ=,TTTξξξBξξξitsufficestoboundthefirstfactorontheright-handsidefromaboveandbelow.ThisalsoyieldsaresultfortheeigenvaluesofB−1A,sincewehaveforthevariableη:=B1/2ξ,ξTAξηTB−1/2AB−1/2η=,ξTBξηTηandthematrixB−1/2AB−1/2possessesthesameeigenvaluesasB−1AbyvirtueofB−1/2(B−1/2AB−1/2)B1/2=B−1A.Here,B1/2isthesymmet-ricpositivedefinitematrixthatsatisfiesB1/2B1/2=B,andB−1/2isitsinverse.TTSinceξAξ/ξBξ=a(v,v)/v,v0anda(v,v)≥αv2≥αv2,10−2(3.140)a(v,v)≤Cv2≤Cminhv2,1K0K∈ThwithagenericconstantC>0(thelastestimateisduetoTheorem3.43,2),itfollowsthatT−2a(v,v)ξAξa(v,v)α≤==≤CminhK,(3.141)v,vξTBξv,vK∈Th00andfromthistheassertion.
TheanalysisoftheeigenvaluesofthemodelprobleminExample2.12showsthattheabove-givenestimatesarenottoopessimistic. 3.8.GeneralDomainsandIsoparametricElements1673.8GeneralDomainsandIsoparametricElementsAllelementsconsideredsofarareboundedbystraightlinesorplanesur-faces.Therefore,onlypolyhedraldomainscanbedecomposedexactlybymeansofatriangulation.Dependingontheapplication,domainswithacurvedboundarymayappear.Withtheavailableelementstheobviouswayofdealingwithsuchdomainsisthefollowing(inthetwo-dimensionalcase):forelementsKthatareclosetotheboundaryputonlythenodesofoneedgeontheboundary∂Ω.Thisimpliesanapproximationerrorforthe)domain,forΩh:=K∈ThK,thereholdsingeneralneitherΩ⊂ΩhnorΩh⊂Ω(seeFigure3.14).BFigure3.14.ΩandΩh.Asthesimplestexample,weconsiderhomogeneousDirichletboundaryconditions,thusV=H1(Ω),onaconvexdomainforwhichthereforeΩ⊂0hΩissatisfied.IfanansatzspaceVhisintroducedasinSection3.3,thenfunctionsdefinedonΩharegenerated.Therefore,thesefunctionsmustbeextendedtoΩinsuchawaythattheyvanishon∂Ω,andconsequently,forthegeneratedfunctionspaceV˜h,V˜h⊂V.ThisissupposedtobedonebyaddingthedomainsBwhoseboundaryconsistsofaboundarypartofsomeelementK∈Thclosetotheboundaryandasubsetof∂ΩtothesetofelementswiththeansatzspaceP(B)={0}.C´ea’slemma(Theorem2.17)canstillbeapplied,sothatforanerrorestimatein·1thequestionofhowtochooseacomparisonelementv∈V˜harises.Theansatzv=I˜h(u),whereI˜h(u)denotestheinterpolationonΩhextendedby0onthedomainsB,isadmissibleonlyforthe(multi-)linearansatz:Onlyinthiscaseareallnodesofanedge“closetotheboundary”locatedon∂Ωandthereforehavehomogeneousdegreesoffreedom,sothatthecontinuityontheseedgesisensured.Forthepresent,letusrestrictourattentiontothiscase,sothatu−I˜h(u)1hastobeestimatedwhereuisthesolutionoftheboundaryvalueproblem.ThetechniquesofSection3.4canbeappliedtoallK∈Th,andbytheconditionsassumedthereaboutthetriangulation,thisyieldsu−uh1≤Cu−Ih(u)1,Ωh+u1,ΩΩh≤Ch|u|2,Ωh+u1,ΩΩh. 1683.FiniteElementMethodsforLinearEllipticProblemsIf∂Ω∈C2,thenwehavetheestimateu1,ΩΩh≤Chu2,Ωforthenewerrorpartduetotheapproximationofthedomain,andthustheconvergencerateispreserved.Alreadyforaquadraticansatzthisisnolongersatisfied,whereonly3/2u−uh1≤Chu3holdsinsteadoftheorderO(h2)ofTheorem3.29(see[31,pp.194ff]).Onemayexpectthatthisdecreaseoftheapproximationqualityarisesonlylocallyclosetotheboundary,however,onemayalsotrytoobtainabetterapproximationofthedomainbyusingcurvedelements.Suchelementscanbedefinedonthebasisofthereferenceelements(K,ˆP,ˆΣ)ofLagrangetypeˆintroducedinSection3.3ifageneralelementisobtainedfromthisonebyanisoparametrictransformation;thatis,chooseandF∈(Pˆ)(3.142)thatisinjectiveandthenK:=F(Kˆ),P:=pˆ◦F−1pˆ∈Pˆ,Σ:=F(ˆa)aˆ∈Σˆ.SincethebijectivityofF:Kˆ→Kisensuredbyrequirement,afiniteelementisthusdefinedintermsof(3.58).Byvirtueoftheuniquesolvabilityoftheinterpolationproblem,Fcanbedefinedbyprescribinga1,...,aL,L=|Σˆ|,andrequiringF(ˆai)=ai,i=1,...,L.However,thisdoesnotingeneralensuretheinjectivity.Since,ontheotherhand,inthegridgenerationprocesselementsarecreatedbydefiningthenodes(seeSection4.1),geometricconditionsabouttheirpositionsthatcharacterizetheinjectivityofFaredesirable.AtypicalcurvedelementthatcanbeusedfortheapproximationoftheboundarycanbegeneratedonthebasisoftheunitsimplexwithPˆ=P2(Kˆ)(seeFigure3.15).â3a32a13ââ23F∈P2(Kˆ)13a
23a1a2â1â12â2a12Figure3.15.Isoparametricelement:quadraticansatzontriangle.Elementswith,ingeneral,onecurvededgeandotherwisestraightedgesthusaresuggestedfortheproblemofboundaryapproximation.Theyare 3.8.GeneralDomainsandIsoparametricElements169combinedwithaffine“quadratictriangles”intheinteriorofthedomain.Subparametricelementscanbegeneratedanalogouslytotheisoparametricelementsif(thecomponentsof)thetransformationsin(3.142)arerestrictedtosomesubspacePˆT⊂Pˆ.IfPˆT=P1(Kˆ),weagainobtaintheaffineequivalentelements.However,isoparametricelementsarealsoimportantif,forinstance,theunitsquareorcubeissupposedtobethereferenceelement.Onlytheisoparametrictransformationallowsfor“general”quadrilateralsandhex-ahedra,respectively,whicharepreferableinanisotropiccases(forinstanceingeneralizationofFigure3.11)tosimplicesduetotheiradaptabilitytolocalcoordinates.Inwhatfollows,letKˆ=[0,1]d,Pˆ=Q(Kˆ).1Ingeneral,sincealsoafiniteelement(inRd−1)isdefinedforeveryfaceSˆofKˆwithPˆ|SˆandΣˆ|Sˆ,the“faces”ofK,thatis,F[Sˆ],arealreadyuniquelydefinedbytherelatednodes.Consequently,ifd=2,theedgesofthegeneralquadrilateralarestraightlines(seeFigure3.16),butifd=3,wehavetoexpectcurvedsurfaces(hyperbolicparaboloids)forageneralhexahedron.a3â4â3a42F∈Q1(Kˆ)
â1â2a1a2Figure3.16.Isoparametricelement:bilinearansatzonrectangle.AgeometriccharacterizationoftheinjectivityofFisstillunknown(toourknowledge)ford=3,butitcanbeeasilyderivedford=2:Letthenodesa1,a2,a3,a4benumberedcounterclockwiseandsupposethattheyarenotonastraightline,andthus(byrearranging)T=conv(a1,a2,a4)formsatrianglesuchthat2vol(T)=det(B)>0.HereFT(ˆx)=Bxˆ+distheaffine-linearmappingthatmapstherefer-−1encetriangleconv(ˆa1,aˆ2,aˆ4)bijectivelytoT.If˜a3:=FT(a3),thenthequadrilateralK˜withtheverticesˆa1,aˆ2,a˜3,aˆ4ismappedbijectivelytoKbyFT.ThetransformationFcanbedecomposedintoF=FT◦FQ,2whereFQ∈Q1(Kˆ)denotesthemappingdefinedbyFQ(ˆai)=ˆai,i=1,2,4,FQ(ˆa3)=˜a3 1703.FiniteElementMethodsforLinearEllipticProblems(seeFigure3.17).x^^2x2a411â4â3â4ã3K^FQ~FTa3KKâ1â2x^â1â2x^a1a11211FFigure3.17.Decompositionofthebilinearisoparametricmapping.Therefore,thebijectivityofFisequivalenttothebijectivityofFQ.Wecharacterizea“uniform”bijectivitywhichisdefinedbydet(DF(ˆx1,xˆ2))=0forthefunctionalmatrixDF(ˆx1,xˆ2):Theorem3.46SupposeQisaquadrilateralwiththeverticesa1,...,a4(numberedcounterclockwise).Then,det(DF(ˆx,xˆ))=0forall(ˆx,xˆ)∈[0,1]2⇐⇒1212det(DF(ˆx,xˆ))>0forall(ˆx,xˆ)∈[0,1]2⇐⇒1212Qisconvexanddoesnotdegenerateintoatriangleorstraightline.Proof:Byvirtueofdet(DF(ˆx1,xˆ2))=det(B)det(DFQ(ˆx1,xˆ2))anddet(B)>0,FcanbereplacedwithFQintheassertion.Sincexˆ1a˜3,1−1FQ(ˆx1,xˆ2)=+xˆ1xˆ2,xˆ2a˜3,2−1itfollowsbysomesimplecalculationsthatdet(DFQ(ˆx1,xˆ2))=1+(˜a3,2−1)ˆx1+(˜a3,1−1)ˆx2isanaffine-linearmappingbecausethequadraticpartsjustcanceleachother.Thismappingassumesitsextremaon[0,1]2atthe4vertices,wherewehavethefollowingvalues:(0,0):1,(1,0):˜a3,2,(0,1):˜a3,1,(1,1):˜a3,1+˜a3,2−1.Auniformsignisthusobtainedifandonlyifthefunctioniseverywherepositive.Thisisthecaseifandonlyifa˜3,1,a˜3,2,a˜3,1+˜a3,2−1>0,whichjustcharacterizestheconvexityandthenondegenerationofK˜.BythetransformationFTthisalsoholdsforK.
3.9.TheMaximumPrincipleforFiniteElementMethods171Accordingtothistheoremitisnotallowedthataquadrilateraldegener-atesintoatriangle(nowwithlinearansatz).Butamorecarefulanalysis[55]showsthatthisdoesnotaffectnegativelythequalityoftheapproximation.Ingeneral,forisoparametricelementswehavethefollowing:Fromthepointofviewofimplementation,onlyslightmodificationshavetobemade:Intheintegrals(3.111),(3.111)transformedtothereferenceelementortheirapproximationbyquadrature(3.120),|detB|hastobereplacedwith|det(DF(ˆx))|(intheintegrand).TheanalysisoftheorderofconvergencecanbedonealongthesamelinesasinSection3.4(and3.6),however,thetransformationrulesfortheintegralsbecomemorecomplex(see[9,pp.237ff.]).3.9TheMaximumPrincipleforFiniteElementMethodsInthissectionmaximumandcomparisionprinciplesthathavebeenintro-ducedforthefinitedifferencemethodareoutlinedforthefiniteelementmethod.Inthecaseoftwo-dimensionaldomainsΩthesituationhasbeenwellinvestigatedforlinearellipticboundaryvalueproblemsofsecondorderandlinearelements.Forhigher-dimensionalproblems(d>2)aswellasothertypesofelements,thecorrespondingassumptionsaremuchmorecomplex,ortheredoesnotnecessarilyexistanymaximumprinciple.Fromnowon,letΩ⊂R2beapolygonallyboundeddomainandletXhdenotethefiniteelementspaceofcontinuous,piecewiselinearfunctionsforaconformingtriangulationThofΩwherethefunctionvaluesinthenodesontheDirichletboundaryΓ3areincludedinthedegreesoffreedom.First,weconsiderthediscretizationdevelopedforthePoissonequation−∆u=fwithf∈L2(Ω).ThealgebraizationofthemethodisdoneaccordingtotheschemedescribedinSection2.4.3.Accordingtothis,firstallnodesinsideΩandonΓ1andΓ2arenumberedconsecutivelyfrom1toanumberM1.Thenodalvaluesuh(ar)forr=1,...,M1arearrangedinthevectoruh.Then,thenodesthatbelongtotheDirichletboundaryarenumberedfromM1+1tosomenumberM1+M2,thecorrespondingnodalvaluesgeneratethevectoruˆh.ThecombinationofuhanduˆhgivesuhMthevectorofallnodalvaluesu˜h=uˆ∈R,M=M1+M2.hThisleadstoalinearsystemofequationsoftheform(1.31)describedinSection1.4:Ahuh=−Aˆhuˆh+fwithA∈RM1,M1,Aˆ∈RM1,M2,u,f∈RM1anduˆ∈RM2.hhhhRecallingthesupportpropertiesofthebasisfunctionsϕi,ϕj∈Xh,weobtainforageneralelementofthe(extended)stiffnessmatrixA˜h:= 1723.FiniteElementMethodsforLinearEllipticProblemsAAˆ∈RM1,Mfollowingtherelationhh

(A˜h)ij=∇ϕj·∇ϕidx=∇ϕj·∇ϕidx.Ωsuppϕi∩suppϕjTherefore,ifi=j,theactualdomainofintegrationconsistsofatmosttwotriangles.Hence,forthepresentitisreasonabletoconsideronlyonetriangleasthedomainofintegration.Lemma3.47SupposeThisaconformingtriangulationofΩ.ThenforanarbitrarytriangleK∈Thwiththeverticesai,aj(i=j),thefollowingrelationholds:
1K∇ϕj·∇ϕidx=−cotαij,K2whereαKdenotestheinteriorangleofKthatisoppositetotheedgewithijtheboundarypointsai,aj.Proof:SupposethetriangleKhastheverticesai,aj,ak(seeFigure3.18).Ontheedgeoppositetothepointaj,wehaveϕj≡0.Therefore,∇ϕjhasthedirectionofanormalvectortothisedgeand—byconsideringinwhichdirectionϕjincreases—theorientationoppositetotheoutwardnormalvectorνki,thatis,∇ϕj=−|∇ϕj|νkiwith|νki|=1.(3.143).ajνjkhj.αKijakνki.aiFigure3.18.NotationfortheproofofLemma3.47.Inordertocalculate|∇ϕj|weusethefollowing:From(3.143)weobtain|∇ϕj|=−∇ϕj·νki;thatis,wehavetocomputeadirectionalderivative.Byvirtueofϕj(aj)=1,wehave0−11∇ϕj·νki==−,hjhj 3.9.MaximumPrincipleforFiniteElementMethods173wherehjdenotestheheightofKwithrespecttotheedgeoppositeaj.Thuswehaveobtainedtherelation1∇ϕj=−νki.hjHencewehavecosαKνki·νjkij∇ϕj·∇ϕi==−.hjhihjhiSince2|K|=h|a−a|=h|a−a|=|a−a||a−a|sinαK,jkiijkkijkijweobtaincosαKij1K1∇ϕj·∇ϕi=−2|ak−ai||aj−ak|=−cotαij,4|K|2|K|sothattheassertionfollowsbyintegration.
Corollary3.48IfKandK
aretwotrianglesofTwhichhaveacommonhedgespannedbythenodesai,aj,then
KK1sin(αij+αij)(A˜h)ij=∇ϕj·∇ϕidx=−KK.K∪K2(sinαij)(sinαij)Proof:Theformulafollowsfromtheadditiontheoremforthecotangentfunction.
Lemma3.47andCorollary3.48arethebasisfortheproofoftheas-sumption(1.32)*inthecaseoftheextendedsystemmatrixA˜h.Indeed,additionalassumptionsaboutthetriangulationTharenecessary:Anglecondition:ForanytwotrianglesofThwithacommonedge,thesumoftheinterioranglesoppositetothisedgedoesnotexceedthevalueπ.IfatrianglehasanedgeontheboundarypartΓ1orΓ2,thentheangleoppositethisedgemustnotbeobtuse.Connectivitycondition:ForeverypairofnodesbothbelongingtoΩ∪Γ1∪Γ2thereexistsapolygonallinebetweenthesetwonodessuchthatthepolygonallineconsistsonlyoftriangleedgeswhoseboundarypointsalsobelongtoΩ∪Γ1∪Γ2(seeFigure3.19).Discussionofassumption(1.32)*:Theproofof(1),(2),(5),(6)*isratherelementary.Forthe“diagonalelements,”

22(Ah)rr=|∇ϕr|dx=|∇ϕr|dx>0,r=1,...,M1,ΩKK⊂suppϕrwhichalreadyis(1).Checkingthesignconditions(2)and(5)forthe“nondiagonalelements”ofA˜hrequirestheanalysisoftwocases: 1743.FiniteElementMethodsforLinearEllipticProblemsFigure3.19.Exampleofanonconnectedtriangulation(Γ3=∂Ω).(i)Forr=1,...,M1ands=1,...,Mwithr=s,thereexisttwotrianglesthathavethecommonverticesar,as.(ii)Thereexistsonlyonetrianglethathasaraswellasasasvertices.Incase(i),Corollary3.48canbeapplied,sinceifK,K
justdenotethetwotriangleswithacommonedgespannedbya,a,then0<αK+αK≤πrsrsrsandthus(A˜h)rs≤0,r=s.Incase(ii),Lemma3.47,duetothepartoftheangleconditionthatreferstotheboundarytriangles,canbeapplieddirectlyyieldingtheassertion.MFurther,sinces=1ϕs=1inΩ,weobtainMM

M(A˜h)rs=∇ϕs·∇ϕrdx=∇ϕs·∇ϕrdx=0.s=1s=1ΩΩs=1Thisis(6)*.Thesignconditionin(3)nowfollowsfrom(6)*and(5),sincewehaveM1MM(Ah)rs=(A˜h)rs−(Aˆh)rs≥0.(3.144)s=1s=1s=M1+1!=0Thedifficultpartoftheproofof(3)consistsinshowingthatatleastoneoftheseinequalities(3.144)issatisfiedstrictly.Thisisequivalenttothefactthatatleastoneelement(Aˆh)rs,r=1,...,M1ands=M1+1,...,M,isnegative,whichcanbeshownintermsofanindirectproofbyusingLemma3.47andCorollary3.48,butisnotdonehereinordertosavespace.Simultaneously,thisalsoprovesthecondition(7).Theremainingcondition(4)*isprovedsimilarly.First,duetothecon-nectivitycondition,theexistenceofgeometricconnectionsbetweenpairsofnodesbypolygonallinesconsistingofedgesisobvious.Itismoredifficulttoprovethatunderallpossibleconnectionsthereexistsonealongwhich 3.9.MaximumPrincipleforFiniteElementMethods175thecorrespondingmatrixelementsdonotvanish.Thiscanbedonebythesametechniqueofproofasusedinthesecondpartof(3),which,however,isnotpresentedhere.Iftheangleconditiongivenaboveisreplacedwithastrongeranglecon-ditioninwhichstretchedandrightanglesareexcluded,thentheproofof(3)and(4)*becomestrivial.Recallingtherelationsmaxuh(x)=max(u˜h)rx∈Ωr∈{1,...,M}andmaxuh(x)=max(uˆh)r,x∈Γ3r∈{M1+1,...,M}whichholdforlinearelements,thefollowingresultcanbederivedfromTheorem1.10.Theorem3.49IfthetriangulationThsatisfiestheangleconditionandtheconnectivitycondition,thenwehavethefollowingestimateforthefiniteelementsolutionuhofthePoissonequationinthespaceoflinearelementsforanonpositiveright-handsidef∈L2(Ω):maxuh(x)≤maxuh(x).x∈Ωx∈Γ3Finally,wemaketworemarksconcerningthecaseofmoregeneraldifferentialequations.Ifanequationwithavariablescalardiffusioncoefficientk:Ω→Riscon-sideredinsteadofthePoissonequation,thentherelationinCorollary3.48losesitspurelygeometriccharacter.Evenifthediffusioncoefficientissupposedtobeelementwiseconstant,thedata-dependentrelation*+1KK(A˜h)ij=−kKcotαij+kKcotαij2wouldarise,wherekKandkKdenotetheconstantrestrictionofktothetrianglesKandK
,respectively.Thecaseofmatrix-valuedcoefficientsK:Ω→Rd,disevenmoreproblematic.Thesecondremarkconcernsdifferentialexpressionsthatalsocontainlower-orderterms,thatis,convectiveandreactiveparts.Ifthediffusiveterm−∇·(K∇u)canbediscretizedinsuchawaythatamaximumprincipleholds,thenthismaximumprincipleispreservedifthediscretiza-tionoftheothertermsleadstomatriceswhose“diagonalelements”arenonnegativeandwhose“nondiagonalelements”arenonpositive.Thesema-trixpropertiesaremuchsimplerthantheconditions(1.32)and(1.32)*.However,satisfyingthesepropertiescausesdifficultiesinspecialcases,e.g.,forconvection-dominatedequations(seeChapter9),unlessadditionalrestrictiveassumptionsaremadeorspecialdiscretizationschemesareused. 4GridGenerationandAPosterioriErrorEstimation4.1GridGenerationAsoneofthefirststeps,theimplementationofthefiniteelementmethod(andalsoofthefinitevolumemethodasdescribedinChapter6)requiresa“geometricdiscretization”ofthedomainΩ.Thispartofafiniteelementprogramisusuallyincludedintheso-calledpreprocessor(seealsoSection2.4.1).Ingeneral,afiniteelementprogramconsistsfurtheroftheintrinsickernel(assemblingofthefinite-dimensionalsystemofalgebraicequations,rearrangementofdata(ifnecessary),solutionofthealgebraicproblem)andthepostprocessor(editingoftheresults,ex-tractionofintermediateresults,preparationforgraphicoutput,aposteriorierrorestimation).4.1.1ClassificationofGridsGridscanbegroupedaccordingtodifferentcriteria:Onecriterionconsidersthegeometricshapeoftheelements(triangles,quadrilaterals,tetrahedra,hexahedra,prisms,pyramids;possiblywithcurvedboundaries).Afurthercriteriondistinguishesthelogicalstructureofthegrid(structuredorun-structuredgrids).Besidetheseroughclasses,inpracticeonecanfindalargenumberofvariantscombininggridsofdifferentclasses(combinedgrids).Astructuredgridinthestrictsenseischaracterizedbyaregulararrange-mentofthegridpoints(nodes),thatis,theconnectivitypatternbetweenneighbouringnodesisidenticaleverywhereintheinteriorofthegrid.The 4.1.GridGeneration177onlyexceptionsofthatpatternmayoccurneartheboundaryofthedomainΩ.TypicalexamplesofstructuredgridsarerectangularCartesiantwo-orthree-dimensionalgridsastheyarealsousedwithintheframeworkofthefinitedifferencemethodsdescribedinChapter1(see,e.g.,Figure1.1).Astructuredgridinthewidersenseisobtainedbytheapplicationofapiecewisesmoothbijectivetransformationtosome“referencegrid”,whichisastructuredgridinthestrictsense.Gridsofthistypearealsocalledlogicallystructured,becauseonlythelogicalstructureoftheconnectivitypatternisfixedintheinteriorofthegrid.However,theedgesorfacesofthegeometricelementsofalogicallystructuredgridarenotnecessarilystraightoreven.Logicallystructuredgridshavetheadvantageofsimpleimplementation,becausethepatternalreadydefinestheneighboursofagivennode.Fur-thermore,thereexistefficientmethodsforthesolutionofthealgebraicsystemresultingfromthediscretization,includingparallelizedresolutionalgorithms.Incontrasttostructuredgrids,unstructuredgridsdonothaveaself-repeatingnodepattern.Moreover,elementsofdifferentgeometrictypecanbecombinedinunstructuredgrids.Unstructuredgridsaresuitabletoolsforthemodellingofcomplexge-ometriesofΩandfortheadjustmentofthegridtothenumericalsolution(localgridadaptation).Inthesubsequentsections,asurveyofafewmethodsforgeneratingunstructuredgridswillbegiven.Methodstoproducestructuredgridscanbefound,forinstance,inthebooks[23]or[33].4.1.2GenerationofSimplicialGridsAsimplicialgridconsistsoftriangles(intwodimensions)ortetrahedra(inthreedimensions).Togeneratesimplicialgrids,thefollowingthreetypesofmethodsarewidelyused:•overlaymethods,•Delaunaytriangulations,•advancingfrontmethods.OverlayMethodsThemethodsofthistypestartwithastructuredgrid(theoverlaygrid)thatcoversthewholedomain.Afterthat,thisbasicgridismodifiedneartheboundarytofittothedomaingeometry.Theso-calledquadtree(intwodimensions)oroctreetechnique(inthreedimensions)formsatypicalexampleofanoverlaymethod,wheretheoverlaygridisarelativelycoarserectangularCartesiantwo-orthree-dimensionalgrid.Thesubstantialpart 1784.GridGenerationandAPosterioriErrorEstimationofthealgorithmconsistsoffittingroutinesforthosepartsofthestartinggridthatarelocatedneartheboundaryandofsimplicialsubdivisionsoftheobtainedgeometricelements.Thefittingproceduresperformrecursivesubdivisionsoftheboundaryrectanglesorrectangularparallelepipedsinsuchawaythatattheendeverygeometricelementcontainsatmostonegeometrydefiningpoint(i.e.,avertexofΩorapointof∂Ω,wherethetypeofboundaryconditionschanges).Finally,theso-calledsmoothingstep,whichoptimizesthegridwithrespecttoacertainregularitycriterion,canbesupplemented;seeSection4.1.4.Typically,gridsgeneratedbyoverlaymethodsareclosetostructuredgridsintheinteriorofthedomain.Neartheboundary,theylosethestructure.Furtherdetailscanbefoundinthereferences[68]and[72].DelaunayTriangulationsThecorealgorithmofthesemethodsgenerates,foragivencloudofisolatedpoints(nodes),atriangulationoftheirconvexhull.Therefore,agridgen-eratorbasedonthisprinciplehastoincludeaprocedureforthegenerationofthispointset(forexample,thepointsresultingfromanoverlaymethod)aswellascertainfittingprocedures(tocover,forexample,nonconvexdomains,too).TheDelaunaytriangulationoftheconvexhullofagivenpointsetinRdischaracterizedbythefollowingproperty(emptyspherecriterion,Fig-ure4.1):Anyopend-ball,theboundaryofwhichcontainsd+1pointsfromthegivenset,doesnotcontainanyotherpointsfromthatset.Thetriangulationcanbegeneratedfromtheso-calledVoronoitesselationofRdforthegivenpointset.Intwodimensions,thisprocedureisdescribedinChapter6,whichdealswithfinitevolumemethods(Section6.2.1).How-Figure4.1.Emptyspherecriterionintwodimensions(d=2). 4.1.GridGeneration179ever,practicalalgorithms([48]or[71])applytheemptyspherecriterionmoredirectly.TheinterestingtheoreticalpropertiesofDelaunaytriangulationsareoneofthereasonsforthe“popularity”ofthismethod.Intwodimensions,theso-calledmax-min-anglepropertyisvalid:AmongalltriangulationsoftheconvexhullGofagivenpointset,theDelaunaytriangulationmaximizestheminimalinteriorangleoveralltriangles.Inthecased=3,thisnicepropertydoesnotremaintrue.Incontrast,evenbadlyshapedelements(theso-calledsliverelements)mayoccur.Afurtherimportantpropertyofatwo-dimensionalDelaunaytriangulationisthatthesumoftwoanglesoppositeaninterioredgeisnotmorethanπ.Forexample,sucharequirementisapartoftheangleconditionformulatedinSection3.9.AdvancingFrontMethodsTheideaofthesemethods,whicharealsoknownintheliterature(see,e.g.,[50],[56],[60],[62])asmovingfrontmethods,istogenerateatriangulationrecursivelyfromadiscretizationofthecurrentboundary.ThemethodsstartwithapartitionoftheboundaryofG0:=Ω.Ford=2,this“initialfront”isapolygonalline,whereasind=3itisatriangulationofacurvedsurface(theso-called“2.5-dimensionaltriangulation”).Themethodcon-sistsofaniterationofthefollowinggeneralstep(Figure4.2):Anelementofthecurrentfront(i.e.,astraight-linesegmentoratriangle)istakenandthen,eithergeneratinganewinnerpointortakinganalreadyexistingpoint,anewsimplexKjthatbelongstoGj−1isdefined.Afterthedataofthenewsimplexaresaved,thesimplexisdeletedfromGj−1.Inthisway,asmallerdomainGjwithanewboundary∂Gj(anew“currentfront”)results.Thegeneralstepisrepeateduntilthecurrentfrontisempty.Of-ten,thegridgenerationprocessissupplementedbytheso-calledsmoothingstep;seeSection4.1.4.KjGj−1GjFigure4.2.Stepjoftheadvancingfrontmethod:ThenewsimplexKjisdeletedfromthedomainGj−1. 1804.GridGenerationandAPosterioriErrorEstimation4.1.3GenerationofQuadrilateralandHexahedralGridsGridsconsistingofquadrilateralsorhexahedracanalsobegeneratedbymeansofoverlaymethods(e.g.,[66])oradvancingfrontmethods(e.g.,[46],[47]).Aninterestingapplicationofsimplicialadvancingfrontmeth-odsinthetwo-dimensionalcaseisgiveninthepaper[73].Themethodisbasedonthesimplefactthatanytwotrianglessharingacommonedgeformaquadrilateral.Obviously,anecessaryconditionforthesuccessofthemethodisthatthetriangulationshouldconsistofanevennumberoftriangles.Unfortunately,thegeneralizationofthemethodtothethree-dimensionalsituationisdifficult,becauseacomparativelylargenumberofadjacenttetrahedrashouldbeunitedtoformahexahedron.MultiblockMethodsThebasicideaofthesemethodsistopartitionthedomainintoasmallnum-berofsubdomains(“blocks”)ofsimpleshape(quadrilaterals,hexahedra,aswellastriangles,tetrahedra,prisms,pyramids,etc.)andthengeneratestructuredorlogicallystructuredgridsintheindividualsubdomains(see,e.g.,[23],[33]).Inmultiblockgrids,specialattentionhastobedevotedtothetreatmentofcommonboundariesofadjacentblocks.Unlessspecialdiscretizationmethodssuchas,forexample,theso-calledmortarfiniteelementmethod(cf.[45])areusedinthissituation,theremaybeaconflictbetweencertaincompatibilityconditionsatthecommonblockinterfaces(toensure,e.g.,thecontinuityofthefiniteelementfunctionsacrosstheinterfaces)ontheonehandandtheoutputdirectivesofanerrorestimationprocedurethatmayadvisetorefineablock-internalgridlocallyontheotherhand.HierarchicallyStructuredGridsThesegridsareafurther,hybridvariantofstructuredandunstructuredgrids,thoughnotyetverywidespread.Startingwithalogicallystructuredgrid,hierarchicallystructuredgridsaregeneratedbyafurtherlogicallystructuredrefinementofcertainsubdomains.Asinmultiblockmethods,theinterfacesbetweenblocksofdifferentrefinementdegreeshavetobetreatedcarefully.CombinedGridsEspeciallyinthree-dimensionalsituations,thegenerationof“purely”hexa-hedralgridsmaybeverydifficultforcomplicatedgeometriesofthedomain.Therefore,theso-calledcombinedgridsthatconsistofhexahedralgridsingeometricallysimplesubdomainsandtetrahedral,prismatic,pyramidal,etc.gridsinmorecriticalsubregionsareused.ChimeraGridsThesegridsarealsocalledoversetgrids(see,e.g.,[51]).Incontrasttothemultiblockgridsdescribedabove,herethedomainiscoveredbyacompar- 4.1.GridGeneration181ativelysmallnumberofdomainsofsimpleshape,andthenstructuredorlogicallystructuredgridsaregeneratedontheindividualdomains.Thatis,acertainoverlappingoftheblocksandthusofthesubgridsisadmitted.4.1.4GridOptimizationManygridgenerationcodesinclude“smoothingalgorithms”thatoptimizethegridwithrespecttocertainregularitycriteria.Intheso-calledr-method(relocationmethod)thenodesareslightlymoved,keepingthelogicalstruc-ture(connectivities)ofthegridfixed.Anotherapproachistoimprovethegridconnectivitiesthemselves.Atypicalexampleforr-methodsisgivenbytheso-calledLaplaciansmoothing(orbarycentricsmoothing),whereanyinnergridpointismovedintothebarycentreofitsneighbours(see[50]).Alocalweightingofselectedneighbourscanalsobeused(weightedbarycentricsmoothing).Fromafor-malpointofview,theapplicationoftheLaplaciansmoothingcorrespondstothesolutionofasystemoflinearalgebraicequationsthatisobtainedfromtheequationsofthearithmetic(orweighted)averageofthenodes.Thematrixofthissystemislargebutsparse.ThestructureofthismatrixisverysimilartotheonethatresultsfromafinitevolumediscretizationofthePoissonequationasdescribedinSection6.2(seethecorrespond-ingspecialcaseof(6.9)).Ingeneral,thereisnoneedtosolvethissystemexactly.Typically,onlyonetothreestepsofasimpleiterativesolver(aspresentedinSection5.1)areperformed.Whenthedomainisalmostcon-vex,Laplaciansmoothingwillproducegoodresults.Itisalsoclearthatforstronglynonconvexdomainsorotherspecialsituations,themethodmayproduceinvalidgrids.Amongthemethodstooptimizethegridconnectivities,theso-called2:1-ruleand,inthetwo-dimensionalcase,theedgeswap(ordiagonalswap,[59])arewellknown.The2:1-ruleisusedwithinthequadtreeoroctreemethodtoreducethedifferenceoftherefinementlevelsbetweenneighbour-ingquadrilateralsorhexahedratoonebymeansofadditionalrefinementsteps;seeFigure4.3.Intheedgeswapmethod,atriangulargridisimproved.Sinceanytwotrianglessharinganedgeformaconvexquadrilateral,themethoddecideswhichofthetwodiagonalsofthequadrilateraloptimizesagivencriterion.Iftheoptimaldiagonaldoesnotcoincidewiththecommonedge,theotherconfigurationwillbetaken;i.e.,theedgewillbeswapped.Finally,itshouldbementionedthatthereexistgridoptimizationmethodsthatdeletenodesorevencompleteelementsfromthegrid.4.1.5GridRefinementAtypicalgridrefinementalgorithmforatriangulargrid,theso-calledred/greenrefinement,haspreviouslybeenintroducedinSection2.4.1.A 1824.GridGenerationandAPosterioriErrorEstimation









Figure4.3.2:1-rule.furtherclassofmethodsisbasedonbisection,thatis,atriangleisdividedbythemedianofanedge.Amethodofbisectionischaracterizedbythenumberofbisectionsusedwithinonerefinementstep(stagenumberofthemethodofbisection)andbythecriterionofhowtoselecttheedgewherethenewnodeistobelocated.Apopularstrategyistotakethelongestofthethreeedges.Thegeneral(recursive)refinementstepforsometriangleKisofthefollowingform:(i)FindthelongestedgeofKandinsertthemedianconnectingthemidpointofthatedgewiththeoppositevertex.(ii)IftheresultingnewnodeisnotavertexofanalreadyexistingtriangleorisnotaboundarypointofthedomainΩ,thentheadjacenttrianglethatsharestherefinededgehastobedivided,too.Sincethelongestedgeoftheadjacenttriangleneednotcoincidewiththecommonedge,theapplicationofthisschemeleadstoanonconformingtriangulation,ingeneral.Toobtainaconformingtriangulation,allnewnodesresultingfromsubstep(i)havetobedetected,andthencertainclosureruleshavetobeapplied.Thered/greenrefinementaswellasthemethodofbisectioncanbegener-alizedtothethree-dimensionalcase.However,sincethenumberofdifferentconfigurationsissignificantlylargerthaninthecased=2,onlyafewillustrativeexampleswillbegiven.Thered/greenrefinementofatetrahedronK(seeFigure4.4)yieldsapartitionofKintoeightsubtetrahedrawiththefollowingproperties:Allverticesofthesubtetrahedracoincideeitherwithverticesorwithedgemid-pointsofK.AtallthefacesofK,thetwo-dimensionalred/greenrefinementschemecanbeobserved.Inadditiontothedifficultiesarisinginthetwo-dimensionalsituation,the(one-stage)bisectionappliedtothelongestedgeofatetrahedronalsoyieldsfacesthatviolatetheconformityconditions.Therefore,theclosurerulesarerathercomplicated,andinpractice,multistage(oftenthree-stage) Exercises183Figure4.4.Representationofthered/greenrefinementofatetrahedron.methodsofbisectionareusedtocircumventthesedifficulties(seeFigure4.5).Figure4.5.Representationofthebisectionofatetrahedron.Gridrefinementmaybenecessaryinthosepartsofthedomainwheretheweaksolutionofthevariationalequationhaslowregularity.Thefigureofthefrontcover(takenfrom[70])showsthedomainforadensity-drivenflowproblem,wheretheinflowandtheoutflowpassthroughverysmall,nearlypoint-sizedsurfaces.Therefinementistheresultofagridadaptationstrategybasedonaposteriorierrorestimators(seeSection4.2).Intime-dependentproblems,wherethosepartsofthegridinwhicharefinementisneededmayalsovary,gridcoarseningisnecessarytolimittheexpense.Asimplegridcoarseningcanbeachieved,forexample,bycancellingformerrefinementstepsinaconformingway.Exercises4.1ForagiventriangleK,thecircumcentrecanbecomputedbyfindingtheintersectionoftheperpendicularbisectorsassociatedwithtwoedges 1844.GridGenerationandAPosterioriErrorEstimationofK.Thiscanbeachievedbysolvingalinearsystemofequationswithrespecttothecoordinatesofthecircumcentre.(a)Givesuchasystem.(b)Howcantheradiusofthecircumcirclebeobtainedfromthissolution?4.2GivenatriangleK,denotebyhithelengthofedgei,i∈{1,2,3}.Provethatthefollowingexpressionequalstheradiusofthecircumcircle(withoutusingthecircumcentre!):h1h2h3.4|K|4.3LetK1,K2betwotrianglessharinganedge.(a)Showtheequivalenceofthefollowingedgeswapcriteria:Anglecriterion:Selectthediagonaloftheso-formedquadrilateralthatmaximizestheminimumofthesixinterioranglesamongthetwoconfigurations.Circlecriterion:Choosethediagonalofthequadrilateralforwhichtheopencircumcirclediskstotheresultingtrianglesdonotcontainanyoftheremainingvertices.(b)Ifα1,α2denotethetwointerioranglesthatarelocatedoppositethecommonedgeofthetrianglesK1andK2,respectively,thenthecirclecriterionstatesthatanedgeswapistobeperformedifα1+α2>π.Provethisassertion.(c)Thecriterionin(b)isnumericallyexpensive.Showthatthefollowingtestisequivalent:[(a1,1−a3,1)(a2,1−a3,1)+(a1,2−a3,2)(a2,2−a3,2)]∗[(a2,1−a4,1)(a1,2−a4,2)−(a1,1−a4,1)(a2,2−a4,2)]<[(a2,1−a4,1)(a1,1−a4,1)+(a2,2−a4,2)(a1,2−a4,2)]∗[(a2,1−a3,1)(a1,2−a3,2)−(a1,1−a3,1)(a2,2−a3,2)].Herea=(a,a)T,i∈{1,2,3},denotetheverticesofatriangleii,1i,2orderedclockwise,anda=(a,a)Tistheremainingvertexof44,14,2thequadrilateral,thepositionofwhichistestedinrelationtothecircumcircledefinedbya1,a2,a3.Hint:Additiontheoremsforthesinfunction. 4.2.APosterioriErrorEstimatesandGridAdaptation1854.2APosterioriErrorEstimatesandGridAdaptationInthepracticalapplicationofdiscretizationmethodstopartialdifferentialequations,animportantquestionishowmuchthecomputedapproximativesolutionuhdeviatesfromtheweaksolutionuofthegivenproblem.Typically,acertainnormoftheerroru−uhistakenasameasureofthisdeviation.Forellipticorparabolicdifferentialequationsofordertwo,acommonnormtoquantifytheerroristheenergynorm(respectivelyanequivalentnorm)ortheL2-norm.Somepracticallyimportantproblemsinvolvetheapproximationoftheso-calledderivedquantitieswhichcanbemathematicallyinterpretedintermsofvaluesofcertainlinearfunctionalsofthesolutionu.Insuchacase,anestimateofthecorrespondingerrorisalsoofinterest.Example4.1J(u)=ν·∇udσ:fluxofuthroughapartoftheboundaryΓ0⊂∂Ω,Γ0J(u)=Ω0udx:integralmeanofuonsomesubdomainΩ0⊂Ω.Inthefollowingwewillconsidersomeestimatesforanorm·oftheerroru−uhandexplainthecorrespondingterminology.Similarstatementsremaintrueifu−uhisreplacedby|J(u)−J(uh)|.Theerrorestimatesgiveninthepreviouschaptersarecharacterizedbythefactthatnoinformationaboutthecomputedsolutionuhisneeded.Estimatesofthistypearecalledapriorierrorestimates.Forexample,consideravariationalequationwithabilinearformthatsatisfies(forsomespaceVsuchthatH1(Ω)⊂V⊂H1(Ω)and·:=·)01theassumptions(2.42),(2.43)andusenumericallypiecewiselinear,con-tinuousfiniteelements.ThenC´ea’slemma(Theorem2.17)togetherwiththeinterpolationerrorestimatefromTheorem3.29impliestheestimateMMu−uh1≤u−Ih(u)1≤Ch,(4.1)ααwheretheconstantCdependsontheweaksolutionuofthevariationalequality.HereChasthespecialform$
%1/2C=C¯|∂αu|2dx(4.2)Ω|α|=2withC>¯0independentofu.Unfortunately,thestructureofthebound(4.2)doesnotallowanimmediatenumericalapplicationof(4.1).ButeveniftheconstantCcouldbeestimatedand(4.1)couldbeusedtodeterminethediscretizationparameterh(maximumdiameterofthetrianglesinTh)foraprescribedtolerance,ingeneralthiswouldleadto 1864.GridGenerationandAPosterioriErrorEstimationagridthatistoofine.Thiscorrespondstoanalgebraicproblemthatistoolarge.Thereasonisthatthedescribedapproachdeterminesaglobalparameter,whereasthetrueerrormeasuremayhavedifferentmagnitudesindifferentregionsofthedomainΩ.Soweshouldaimaterrorestimatesoftypeu−uh≤Dη(4.3)orD1η≤u−uh≤D2η,(4.4)wheretheconstantsD,D1,D2>0donotdependonthediscretizationparametersand$%1/2η=η2.(4.5)KK∈ThHerethequantitiesηKshouldbecomputableusingonlythedata—includingpossiblyuh|K—whichareknownontheparticularelementK.Iftheboundsη(orthetermsηK,respectively)in(4.3)(respectively(4.4))dependonuh,i.e.,theycanbeevaluatedonlyifuhisknown,thentheyarecalled(local)aposteriorierrorestimatorsinthewidersense.Oftentheboundsalsodependontheweaksolutionuofthevariationalequality,soinfact,theycannotbeevaluatedimmediately.Insuchacasetheyshouldbereplacedbycomputablequantitiesthatdonotdependonuinadirectway.So,iftheboundscanbeevaluatedwithoutknowingubutusingpossiblyuh,thentheyarecalled(local)aposteriorierrorestimatorsinthestrictsense.Inequalitiesoftheform(4.3)guarantee,foragiventoleranceε>0,thattheinequalityη≤εimpliesthattheerrormeasuredoesnotexceedεuptoamultiplicativeconstant.Inthissensetheerrorestimatorηiscalledreliable.Now,ifthecomputedapproximativesolutionuhissufficientlypreciseinthedescribedsense,thenthecomputationcanbefinished.Ifuhissuchthatη>ε,thenthequestionofhowtomodifythediscretizationinordertoachievethetoleranceor,ifthecomputerresourcesarenearlyexhausted,howtominimizetheovershootingofη,arises.Thatis,theinformationgivenbytheevaluationoftheboundshastobeusedtoadaptthediscretizationandthentoperformanewrunofthesolutionprocess.Atypicalmodificationistorefineortocoarsenthegrid.Errorestimatorsmayoverestimatetherealerrormeasuresignificantly;thusagridadaptationprocedurebasedonsuchanerrorestimategener-atesagridthatistoofine,andconsequently,thecorrespondingalgebraicproblemistoolarge.Thiseffectcanbereducedorevenavoidediftheerrorestimatorsatisfiesatwo-sidedinequalitylike(4.4).ThentheratioD2/D1isameasureoftheefficiencyoftheerrorestimator. 4.2.APosterioriErrorEstimatesandGridAdaptation187Anerrorestimatorηiscalledasymptoticallyexactifforanarbitrarycon-vergentsequenceofapproximations{uh}withu−uh→0thefollowinglimitisvalid:η→1.u−uhUsually,aposteriorierrorestimatorsaredesignedforawell-definedclassofboundaryorinitial-boundaryvalueproblems.Withinagivenclassofproblems,thequestionregardingthesensitivityoftheconstantsDin(4.3)orD1,D2in(4.4),withrespecttotheparticulardataoftheproblem(e.g.,coefficients,inhomogeneities,geometryofthedomain,gridgeometry,...),arises.Ifthisdependenceofthedataisnotcrucial,thentheerrorestimatoriscalledrobustwithinthisclass.GridAdaptationLetusassumethatthelocalerrorestimatorsηKcomposinganefficienter-rorestimatorηforanapproximatesolutionuhonsomegridThreallyreflecttheerrorontheelementKandthatthislocalerrorcanbeimprovedbyarefinementofK(e.g.,followingtheprinciplesofSection4.1.5).Thenthefollowinggridadaptationstrategiescanbeapplieduntilthegiventoleranceεisreachedorthecomputerresourcesareexhausted.Equidistributionstrategy:Theobjectiveofthegridadaptation(refinementorcoarseningofelements)istogetanewgridTnewsuchthatthehlocalerrorestimatorsηnewforthisnewgridtakeoneandthesameKvalueforallelementsK∈Tnew;thatis(cf.(4.5))hnewεnewηK≈"newforallK∈Th.|T|hSincethenumberofelementsofthenewgridenterstheright-handsideofthiscriterion,thestrategyisanimplicitmethod.Inpracticaluse,itisapproximatediteratively.Cut-offstrategy:Givenaparameterκ∈(0,1),athresholdvalueκηisdefined.ThentheelementsKwithηK>κηwillberefined.Reductionstrategy:Givenaparameterκ∈(0,1),anauxiliarytoler-anceεη:=κηisdefined.Thenacoupleofstepsfollowingtheequidistributionstrategywiththetoleranceεηareperformed.Inpractice,theequidistributionstrategymayperformcomparativelyslowlyandthusmayreducetheefficiencyofthecompletesolutionprocess.Thecut-offmethoddoesnotallowgridcoarsening.Itisrathersensitivetothechoiceoftheparameterκ.Amongallthreestrategies,thereductionmethodrepresentsthebestcompromise. 1884.GridGenerationandAPosterioriErrorEstimationDesignofAPosterioriErrorEstimatorsInthefollowing,threebasicprinciplesofthedesignofaposteriorierrorestimatorswillbedescribed.Inordertoillustratetheunderlyingideasandtoavoidunnecessarytechnicaldifficulties,amodelproblemwillbetreated:Consideradiffusion-reactionequationonapolygonallyboundeddomainΩ⊂R2withhomogeneousDirichletboundaryconditions−∆u+ru=finΩ,u=0on∂Ω,wheref∈L2(Ω)andr∈C(Ω)withr(x)≥0forallx∈Ω.Theproblemisdiscretizedusingpiecewiselinear,continuousfiniteelementfunctionsasdescribedinSection2.2.Settinga(u,v):=(∇u·∇v+ruv)dxforu,v∈V:=H1(Ω),wehaveΩ0thefollowingvariational(weak)formulation:Findu∈Vsuchthata(u,v)=f,v0forallv∈V.Thecorrespondingfiniteelementmethodreadsasfollows:Finduh∈Vhsuchthata(uh,vh)=f,vh0forallvh∈Vh.ResidualErrorEstimatorsSimilartothederivationoftheapriorierrorestimateintheproofofC´ea’slemma(Theorem2.17),theV-ellipticityofa(2.43)impliesthat2αu−uh1≤a(u−uh,u−uh).Withoutlossofgeneralitywemaysupposeu−uh∈V{0},hence1a(u−uh,u−uh)1a(u−uh,v)u−uh1≤≤sup.(4.6)αu−uh1αv∈Vv1Weobservethattheterma(u−uh,v)=a(u,v)−a(uh,v)=f,v0−a(uh,v)(4.7)istheresidualofthevariationalequation;i.e.,theright-handsideofin-equality(4.6)canbeinterpretedasacertainnormofthevariationalresidual.Inanextstep,thevariationalresidualwillbesplitintolocaltermsaccordingtothegivengrid,andthesetermsaretransformedbymeansofintegrationbyparts.Forarbitraryv∈V,from(4.7)itfollowsthat

a(u−uh,v)=fvdx−(∇uh·∇v+ruhv)dxK∈ThKK

=[f−(−∆uh+ruh)]vdx−ν·∇uhvdσ.K∈ThK∂K 4.2.APosterioriErrorEstimatesandGridAdaptation189ThefirstfactorintheintegralsovertheelementsKistheclassicalelementwiseresidualofthedifferentialequation:rK(uh):=[f−(−∆uh+ruh)]KAllquantitiesenteringrK(uh)areknown.Inthecaseconsideredhereweevenhave−∆uh=0onK,hencerK(uh)=[f−ruh]K.TheintegralsovertheboundaryoftheelementsKarefurthersplitintoasumovertheintegralsalongtheelementedgesE⊂∂K:

ν·∇uhvdσ=ν·∇uhvdσ.∂KEE⊂∂KSincev=0on∂Ω,onlytheintegralsalongedgeslyinginΩcontributetothesum.DenotingbyEhthesetofallinterioredgesofallelementsK∈ThandassigningafixedunitnormalνEtoanyofthoseedges,weseethatinthesummationofthesplitboundaryintegralsoverallK∈ThthereoccurexactlytwointegralsalongoneandthesameedgeE∈Eh.Thisobservationresultsintherelation

ν·∇uhvdσ=[νE·∇uh]Evdσ,K∈Th∂KE∈EhEwhere,forapiecewisecontinuousfunctionw:Ω→R,theterm[w]E(x):=limw(x+νE)−limw(x−νE),x∈E,→+0→+0denotesthejumpofthefunctionwacrosstheedgeE.IfwisthenormalderivativeofuhinthefixeddirectionνE,i.e.,w=νE·∇uh,thenitsjumpdoesnotdependontheparticularorientationofνE(seeExercise4.6).Insummary,wehavethefollowingrelation:

a(u−uh,v)=rK(uh)vdx−[νE·∇uh]Evdσ.KEK∈ThE∈EhUsingtheerrorequation(2.39),weobtainforanarbitraryelementvh∈Vhthefundamentalidentitya(u−uh,v)=a(u−uh,v−vh)
=rK(uh)(v−vh)dxKK∈Th
−[νE·∇uh]E(v−vh)dσ,EE∈Ehwhichisthestartingpointfortheconstructionoffurtherestimates. 1904.GridGenerationandAPosterioriErrorEstimationKEFigure4.6.Thetriangularneighbourhoods∆(K)(left)and∆(E)(right).FirstweseethattheCauchy–Schwarzinequalityimmediatelyimpliesa(u−uh,v−vh)≤rK(uh)0,Kv−vh0,KK∈Th(4.8)+[νE·∇uh]E0,Ev−vh0,E.E∈EhTogetthisboundassmallaspossible,thefunctionvh∈Vhischosensuchthattheelementv∈VisapproximatedadequatelyinbothspacesL2(K)andL2(E).Onesuggestionistheuseofaninterpolatingfunctionaccordingto(2.47).However,sinceV∈C(Ω),thisinterpolantisnotdefined.There-foreotherapproximationprocedureshavetobeapplied.Roughlyspeaking,suitableapproximationprinciples,duetoCl´ement[52]orScottandZhang[67],arebasedontakingcertainlocalintegralmeans.However,atthisplacewecannotgofurtherintothesedetailsandrefertothecitedliterature.Infact,forourpurposesitisimportantonlythatsuchapproximationsexist.Theirparticulardesignisofminorinterest.Wewillformulatetherelevantfactsasalemma.Todoso,weneedsomeadditionalnotation(seeFigure4.6):)triangularneighbourhoodofatriangleK:∆(K):=K
,K:K∩K=∅)triangularneighbourhoodofanedgeE:∆(E):=K
.K:K∩E=∅Thus∆(K)consistsoftheunionofthesupportsofthosenodalbasisfunctionsthatareassociatedwiththeverticesofK,whereas∆(E)isformedbytheunionofthosenodalbasisfunctionsthatareassociatedwiththeboundarypointsofE.Furthermore,thelengthoftheedgeEisdenotedbyhE:=|E|.Lemma4.2Letaregularfamily(Th)oftriangulationsofthedomainΩbegiven.Thenforanyv∈VthereexistsanelementQhv∈VhsuchthatforalltrianglesK∈ThandalledgesE∈Ehthefollowingestimatesarevalid:v−Qhv0,K≤ChK|v|1,∆(K), 4.2.APosterioriErrorEstimatesandGridAdaptation191"v−Qhv0,E≤ChE|v|1,∆(E),wheretheconstantC>0dependsonlyonthefamilyoftriangulations.Now,settingvh=Qhvin(4.8),thediscreteCauchy-Schwarzinequalityyieldsa(u−uh,v)≤ChKrK(uh)0,K|v|1,∆(K)K∈Th"+ChE[νE·∇uh]E0,E|v|1,∆(E)E∈Eh$%1/2$%1/2222≤ChKrK(uh)0,K|v|1,∆(K)K∈ThK∈Th$%1/2$%1/22+Ch[ν·∇u]|v|2.EEhE0,E1,∆(E)E∈EhE∈EhAdetailedinvestigationofthetwosecondfactorsshowsthatwecandecomposetheintegralsover∆(K),∆(E),accordingto



...=...,...=....∆(K)K∆(E)KK⊂∆(K)K⊂∆(E)ThisleadstoarepeatedsummationoftheintegralsoverthesingleelementsK.However,duetotheregularityofthefamilyoftriangulations,themul-tiplicityofthesesummationsisboundedindependentlyoftheparticulartriangulation(see(3.93)).Sowearriveattheestimates|v|2≤C|v|2,|v|2≤C|v|2.1,∆(K)11,∆(E)1K∈ThE∈Eh"Usingtheinequalitya+b≤2(a2+b2)fora,b∈R,wegeta(u−uh,v)$%1/2≤Ch2r(u)2+h[ν·∇u]2|v|.KKh0,KEEhE0,E1K∈ThE∈EhFinally,(4.6)yieldsu−u≤Dηwithη2:=η2h1KK∈Thand22212ηK:=hKf−ruh0,K+hE[νE·∇uh]E0,E.(4.9)2E⊂∂K∂Ω 1924.GridGenerationandAPosterioriErrorEstimationHerewehavetakenintoaccountthatinthetransformationoftheedgesum...intothedoublesum...E∈EhK∈ThE⊂∂K∂Ωthelattersumsupeveryinterioredgetwice.Insummary,wehaveobtainedanaposteriorierrorestimateoftheform(4.3).Bymeansofrefinedargumentsitisalsopossibletoderivelowerboundsforu−uh1.Fordetails,werefertotheliterature,forexample[35].ErrorEstimationbyGradientRecoveryIfweareinterestedinanestimateoftheerroru−u∈V=H1(Ω)h0measuredintheH1-orenergynorm·,thisproblemcanbesimplifiedbymeansofthefactthatbothnormsareequivalentonVtotheH1-seminorm
1/22|u−uh|1=|∇u−∇uh|dx=:∇u−∇uh0.ΩThisisasimpleconsequenceofthedefinitionsandthePoincar´einequality(seeTheorem2.18).Thatis,thereexistconstantsC1,C2>0independentofhsuchthatC1|u−uh|1≤u−uh≤C2|u−uh|1(4.10)(cf.Exercise4.8).Consequently,∇uremainstheonlyunknownquantityintheerrorbound.TheideaoferrorestimationbymeansofgradientrecoveryistoreplacetheunknowngradientoftheweaksolutionubyasuitablequantityRhuhthatiscomputablefromtheapproximativesolutionuhatmoderateex-pense.Apopularexampleofsuchatechniqueistheso-calledZ2estimate.Herewewilldescribeasimpleversionofit.FurtherapplicationscanbefoundintheoriginalpapersbyZienkiewiczandZhu,e.g.,[74].Similartothenotationintroducedintheprecedingsubsection,foragivennodeatheset#∆(a):=K
K:a∈∂Kdenotesthetriangularneighbourhoodofa(seeFigure4.7).Thissetco-incideswiththesupportofthepiecewiselinear,continuousbasisfunctionassociatedwiththatnode.Thegradient∇uhofapiecewiselinear,continuousfiniteelementfunctionuhisconstantoneverytriangleK.ThissuggeststhatatanynodeaofthetriangulationThwedefinetheaverageRhuh(a)ofthevaluesofthegradientsonthosetrianglessharingthevertexa:1Rhuh(a):=∇uh|K|K|.|∆(a)|K⊂∆(a) 4.2.APosterioriErrorEstimatesandGridAdaptation193aFigure4.7.Thetriangularneighbourhood∆(a).InterpolatingthetwocomponentsofthesenodalvaluesofRhuhseparatelyinVh,wegetarecoveryoperatorRh:Vh→Vh×Vh.Nowalocalerrorestimatorcanbedefinedbythesimplerestrictionofthequantityη:=Rhuh−∇uh0ontoasingleelementK:ηK:=Rhuh−∇uh0,K.AniceinsightintothepropertiesofthislocalestimatorwasgivenbyRodr´ıguez([64],seealso[35]),whocompareditwiththecorrespondingresidualestimator(4.9).Namely,neglectingintheresidualestimatorjusttheresidualpart,i.e.,setting21222η˜K:=hE[νE·∇uh]E0,Eandη˜:=η˜K,2E⊂∂K∂ΩK∈Ththenthefollowingresultistrue:Theorem4.3Thereexisttwoconstantsc1,c2>0dependingonlyonthefamilyoftriangulationssuchthatc1η˜≤η≤c2η.˜ThemotivationforthemethodofgradientrecoveryistobeseeninthefactthatRhuhpossessesspecialconvergenceproperties.Namely,undercertainassumptionstherecoveredgradientRhuhconvergesasymptoti-callyto∇ufasterthan∇uhdoes.InsuchacaseRhuhissaidtobeasuperconvergentapproximationto∇u.Ifsuperconvergenceholds,thesimpledecomposition∇u−∇uh=Rhuh−∇uh+∇u−Rhuhdemonstratesthatthefirstdifferenceontheright-handsiderepresentstheasymptoticallydominating,computablepartofthegradienterror∇u−∇uh.Inotherwords,ifwecoulddefine,fortheclassofproblemsunderconsideration,asuperconvergentgradientrecoveryRhuhthatiscom-putablewithmoderateexpense,thenthequantitiesηKandηdefinedabovemayserveasatoolforaposteriorierrorestimation. 1944.GridGenerationandAPosterioriErrorEstimationUnfortunately,suchsuperconvergencepropertiesarevalidonlyunderratherrestrictiveassumptions(especiallywithrespecttothegridandtotheregularityoftheweaksolution).Thusitisdifficulttoobtainafullmath-ematicalfoundationinpractice.Nevertheless,gradientrecoveryisoftenappliedandyieldssatisfactoryresultsinmanysituations.Thefollowingexample,whichisduetoRepin[63],showsthatarecoveredgradientdoesnothavetoreflecttherealbehaviouroftheerror.Example4.4Considerthefollowingboundaryvalueproblemford=1andΩ=(0,1):−u

=finΩ,u(0)=u(1)−1=0.Iffisconstant,theexactsolutionreadsu(x)=x(2+(1−x)f)/2.Supposewehavefoundthefunctionvh=xasanapproximatesolution.ForanarbitrarypartitionofΩintosubintervals,thisfunctionispiecewiselinearanditsatisfiestheboundaryconditionsformulatedabove.NowletRhbeanarbitrarygradientrecoveryoperatorthatisabletoreproduceatleastconstants.Sincev
=1,wehavev
−Rv=0,whereastherealerrorishhhhv
−u
=(x−1)f.h2Aninterpretationofthiseffectisthatthefunctionvhdoesnotsolvethecorrespondingdiscrete(Galerkin)equations.Butthispropertyofuhisusedfortheproofofsuperconvergence.Thispropertyalsoplaysanimportantroleinthederivationoftheresidualerrorestimates,becausetheerrorequationisusedtherein.Dual-WeightedResidualErrorEstimatorsTheaforementionedaposteriorierrorestimateshavetwodisadvantages:Ontheonehand,certainglobalconstants,whicharenotknowningeneral,arepartofthebounds.Typicalexamplesareα−1in(4.6)andtheconstantsC1,C2intheequivalencerelation(4.10).Ontheotherhand,weobtained√scalingfactorslikehKandhEsimplybyusingaparticularapproximationoperator.Inthefollowing,wewilloutlineamethodthatattemptstocircumventthesedrawbacks.Itisespeciallyappropriatefortheestimationoferrorsoffunctionalsdependinglinearlyonthesolution.SoletJ:V→Rdenotealinear,continuousfunctional.Weareinterestedinanestimateof|J(u)−J(uh)|.Therefore,thefollowingauxiliarydualproblemisconsidered:Findw∈Vsuchthata(v,w)=J(v)forallv∈V.Takingv=u−uh,wegetimmediatelyJ(u)−J(uh)=J(u−uh)=a(u−uh,w).Ifwh∈Vhisanarbitraryelement,theerrorequation(2.39)yieldsJ(u)−J(uh)=a(u−uh,w−wh). 4.2.APosterioriErrorEstimatesandGridAdaptation195Obviously,theright-handsideisofthesamestructureasinthederivationoftheestimate(4.8).Consequently,byusingthesameargumentsitfollowsthat|J(u)−J(uh)|≤rK(uh)0,Kw−wh0,KK∈Th+[νE·∇uh]E0,Ew−wh0,E.E∈EhIncontrasttothepreviousapproaches,herethenormsofw−whwillnotbetheoreticallyanalyzedbutnumericallyapproximated.Thiscanbedonebyanapproximationofthedualsolutionw.Thereareseveral(moreorlessheuristic)waystodothis.(1)Estimationoftheapproximationerror:Here,thenormsofw−whareestimatedasinthecaseofresidualerrorestimators.SincetheresultdependsontheunknownH1-seminormofw,whichisequiva-lenttotheL2-normof∇w,thefiniteelementsolutionw∈Vofthehhauxiliaryproblemisusedtoapproximate∇w.Itisagreatdisadvan-tageofthisapproachthatagainglobalconstantsenterinthefinalestimatethroughtheestimationoftheapproximationerror.Further-more,thediscreteauxiliaryproblemisofsimilarcomplexitytothatoftheoriginaldiscreteproblem.(2)Higher-orderdiscretizationsoftheauxiliaryproblem:Theauxiliaryproblemissolvednumericallybyusingamethodthatismoreaccu-ratethantheoriginalmethodtodetermineasolutioninVh.Thenwisreplacedbythatsolutionandwh∈Vhbyaninterpolantofthatsolution.Unfortunately,sincethediscreteauxiliaryproblemisofcomparativelylargedimension,thisapproachisratherexpensive.(3)Approximationbymeansofhigher-orderrecovery:Thismethodworkssimilarlytotheapproachdescribedintheprevioussubsection;wisreplacedbyanelementthatisrecoveredfromthefiniteelementsolutionwh∈Vhoftheauxiliaryproblem.Therecoveredelementapproximateswwithhigherorderinbothnormsthanwhdoes.Thismethodexhibitstwoproblems:Ontheonehand,theauxiliaryprob-lemhastobesolvednumerically,andontheotherhand,ensuringthecorrespondingsuperconvergencepropertiesmaybedifficult.Attheendofthissectionwewanttomentionhowthemethodcouldbeusedtoestimatecertainnormsoftheerror.Inthecasewherethenormsareinducedbyparticularscalarproducts,thereisasimple,formalway.Forexample,fortheL2-normwehaveu−uh,u−uh0u−uh0=.u−uh0 1964.GridGenerationandAPosterioriErrorEstimationKeepinguanduhfixed,wegetwiththedefinitionv,u−uh0J(v):=u−uh0alinear,continuousfunctionalJ:H1(Ω)→RsuchthatJ(u)−J(u)=hu−uh0.ThepracticaldifficultyofthisapproachconsistsinthefactthattobeabletofindthesolutionwoftheauxiliaryproblemwehavetoknowthevaluesofJ,buttheydependontheunknownelementu−uh.Theideaofapproximatingthesevaluesimmediatelyimpliestwoproblems:Thereisadditionalexpense,andtheinfluenceoftheapproximationqualityontheaccuracyoftheobtainedboundshastobeanalyzed.Exercises4.4LetΩ⊂R2beaboundeddomainwithapolygonal,Lipschitzcon-tinuousboundaryandV:=H1(Ω).NowconsideraV-elliptic,continuous0bilinearformaandacontinuouslinearformb.Theproblemu∈V:a(u,v)=b(v)forallv∈Visdiscretizedusingpiecewiselinear,continuousfiniteelements.IfEide-notesthesupportofthenodalbasisfunctionsofVhassociatedwiththevertexai,showthattheabstractlocalerrorindicatorsa(e,v)ηi:=supv∈H1(Ei)v0canbeestimatedbymeansofthesolutionse∈H1(E)ofthelocali0iboundaryvalueproblemse∈H1(E):a(e,v)=b(v)−a(u,v)forallv∈H1(E)i0iih0iasfollows(Mandαdenotetheconstantsappearinginthecontinuityandellipticityconditionsona):αei≤ηi≤Mei.Ifnecessary,theelementsofH1(E)areextendedbyzerotothewhole0idomainΩ.4.5Alinearpolynomialonsometriangleisuniquelydefinedeitherbyitsvaluesattheverticesorbyitsvaluesattheedgemidpoints.Forafixedtriangulationofapolygonallybounded,simplyconnecteddomainΩ⊂R2,therecanbedefinedtwofiniteelementspacesbyidentifyingcommondegreesoffreedomofadjacenttriangles. Exercises197(a)Showthatthedimensionofthespacedefinedbythedegreesoffree-domlocatedattheverticesislessthanthedimensionoftheotherspace(providedthatthetriangulationconsistsofmorethanonetriangle).(b)Howcanoneexplainthis“lossofdegreesoffreedom”?4.6DenotebyTatriangulationofthedomainΩ⊂Rd.Showthatforhafunctionv:Ω→Rthatiscontinuouslydifferentiableoneachelementthejump[νE·∇v]EofthenormalderivativeofvacrossanelementedgeEdoesnotdependontheorientationofthenormalνE.4.7Letaregularfamilyoftriangulations(T)ofadomainΩ⊂R2behgiven.ShowthatthereexistconstantsC>0thatdependonlyonthefamily(Th)suchthat|v|2≤Cv2forallv∈L2(Ω),0,∆(K)0K∈Th|v|2≤Cv2forallv∈L2(Ω).0,∆(E)0E∈Eh4.8LetΩ⊂Rdbeaboundeddomain.ShowthatthereareconstantsC,C>0suchthatforallv∈H1(Ω),120C1|v|1≤v1≤C2|v|1. 5IterativeMethodsforSystemsofLinearEquationsWeconsideragainthesystemoflinearequationsAx=b(5.1)withnonsingularmatrixA∈Rm,m,right-handsideb∈Rm,andsolutionx∈Rm.AsshowninChapters2and3,suchsystemsofequationsarisefromfiniteelementdiscretizationsofellipticboundaryvalueproblems.ThematrixAisthestiffnessmatrixandthussparse,ascanbeseenfrom(2.37).Asparsematrixisvaguelyamatrixwithsomanyvanishingelementsthatusingthisstructureinthesolutionof(5.1)isadvantageous.Takingadvan-tageofabandorhullstructurewasdiscussedinSection2.5.Moreprecisely,if(5.1)representsafiniteelementdiscretization,thenitisnotsufficienttoknowthepropertiesofthesolutionmethodforafixedm.Itisonthecon-trarynecessarytostudyasequenceofproblemswithgrowingdimensionm,asitappearsbytherefinementofatriangulation.Inthestrictsenseweunderstandbythenotionsparsematricesasequenceofmatricesinwhichthenumberofnonzeroelementsperrowisboundedindependentlyofthedimension.Thisisthecaseforthestiffnessmatricesdueto(2.37)iftheun-derlyingsequenceoftriangulationsisregularinthesenseofDefinition3.28,forexample.Infiniteelementdiscretizationsoftime-dependentproblems(Chapter7)aswellasinfinitevolumediscretizations(Chapter6)systemsofequationsofequalpropertiesarise,sothatthefollowingconsiderationscanbealsoappliedthere.Thedescribedmatrixstructureisbestappliediniterativemethodsthathavetheoperationmatrix×vectorasanessentialmodule,whereeitherthesystemmatrixAoramatrixofsimilarstructurederivedfromitis 5.IterativeMethodsforSystemsofLinearEquations199concerned.Ifthematrixissparseinthestrictsense,thenO(m)elementaryoperationsarenecessary.Inparticular,list-orientedstorageschemescanbeofuse,aspointedoutinSection2.5.Theeffortfortheapproximativesolutionof(5.1)byaniterativemethodisdeterminedbythenumberofelementaryoperationsperiterationstepandthenumberofiterationskthatarenecessaryinordertoreachthedesiredrelativeerrorlevelε>0,i.e.,tomeetthedemandx(k)−x≤εx(0)−x.(5.2)Herex(k)isthesequenceofiteratesfortheinitialvaluex(0),·afixedknorminRm,andx=A−1btheexactsolutionof(5.1).Forallmethodstobediscussedwewillhavelinearconvergenceofthekindx(k)−x≤kx(0)−x(5.3)withacontractionnumberwith0<<1,whichingeneraldependsonthedimensionm.Tosatisfy(5.2),kiterationsarethussufficient,with211k≥lnln.(5.4)εThecomputationaleffortofamethodobviouslydependsonthesizeofε,althoughthiswillbeseenasfixedandonlythedependenceonthedimen-sionmisconsidered:oftenεwillbeomittedinthecorrespondingLandau’ssymbols.Themethodsdifferthereforebytheirconvergencebehaviour,de-scribedbythecontractionnumberandespeciallybyitsdependenceonm(forspecificclassesofmatricesandboundaryvalueproblems).Amethodis(asymptotically)optimalifthecontractionnumbersareboundedindependentlyofm:(m)≤<1.(5.5)InthiscasethetotaleffortforasparsematrixisO(m)elementaryopera-tions,asforamatrix×vectorstep.Ofcourse,foramoreexactcomparison,thecorrespondingconstants,whichalsoreflecttheeffortofaniterationstep,havetobeexactlyestimated.Whiledirectmethodssolvethesystemofequations(5.1)withmachineprecision,provideditissolvableinastablemanner,onecanfreelychoosetheaccuracywithiterativemethods.If(5.1)isgeneratedbythediscretiza-tionofaboundaryvalueproblem,itisrecommendedtosolveitonlywiththataccuracywithwhich(5.1)approximatestheboundaryvalueprob-lem.Asymptoticstatementsheretohave,amongothers,beendevelopedin(3.89),(7.129)andgiveanestimationoftheapproximationerrorbyChα,withconstantsC,α>0,wherebyhisthemeshsizeofthecorrespondingtriangulation.Sincetheconstantsintheseestimatesareusuallyunknown,theerrorlevelcanbeadaptedonlyasymptoticallyinm,inordertogain 2005.IterativeMethodsforSystemsofLinearEquationsanalgorithmicerrorofequalasymptoticscomparedtotheerrorofap-proximation.Althoughthiscontradictstheabove-describedpointofviewofaconstanterrorlevel,itdoesnotalteranythinginthecomparisonofthemethods:Therespectiveeffortalwayshastobemultipliedbyafac-torO(lnm)ifindspacedimensionsm∼h−disvalid,andtherelationsbetweenthemethodscomparedremainthesame.Furthermore,thechoiceoftheerrorlevelεwillbeinfluencedbythequalityoftheinitialiterate.Generally,statementsabouttheinitialiterateareonlypossibleforspecialsituations:Forparabolicinitialboundaryvalueproblems(Chapter7)andaone-steptimediscretizationitisrecommendedtousetheapproximationoftheoldtimelevelasinitialiterate.Inthecaseofahierarchyofspacediscretizations,anestediterationispossible(Section5.6),wheretheinitialiterateswillnaturallyresult.5.1LinearStationaryIterativeMethods5.1.1GeneralTheoryWebeginwiththestudyofthefollowingclassofaffine-lineariterationfunctions,Φ(x):=Mx+Nb,(5.6)withmatricesM,N∈Rm,mtobespecifiedlater.BymeansofΦaniter-ationsequencex(0),x(1),x(2),...isdefinedthroughafixed-pointiterationx(k+1):=Φx(k),k=0,1,...,(5.7)fromaninitialapproximationx(0).Methodsofthiskindarecalledlinearstationary,becauseoftheirform(5.6)withafixediterationmatrixM.ThefunctionΦ:Rm→Rmiscontinuous,sothatincaseofconvergenceofx(k)fork→∞,forthelimitxwehavex=Φ(x)=Mx+Nb.Inordertoachievethatthefixed-pointiterationdefinedby(5.6)iscon-sistentwithAx=b,i.e.,eachsolutionof(5.1)isalsoafixedpoint,wemustrequire−1−1mAb=MAb+Nbforarbitraryb∈R,i.e.,A−1=MA−1+N,andthusI=M+NA.(5.8)Ontheotherhand,ifNisnonsingular,whichwillalwaysbethecaseinthefollowing,then(5.8)alsoimpliesthatafixedpointof(5.6)solvesthesystemofequations. 5.1.LinearStationaryIterativeMethods201Assumingthevalidityof(5.8),thefixed-pointiterationfor(5.6)canalsobewrittenasx(k+1)=x(k)−NAx(k)−b,(5.9)becauseMx(k)+Nb=(I−NA)x(k)+Nb.IfNisnonsingular,wehaveadditionallyanequivalentformgivenbyWx(k+1)−x(k)=−Ax(k)−b(5.10)withW:=N−1.Thecorrectionx(k+1)−x(k)forx(k)isgivenbytheresidualg(k):=Ax(k)−bthrough(5.9)or(5.10),possiblybysolvingasystemofequations.Inordertocompetewiththedirectmethod,thesolutionof(5.10)shouldrequireoneorderinmfewerelementaryoperations.FordensematricesnomoreoperationsthanO(m2)shouldbenecessaryasarealreadynecessaryforthecalculationofg(k).Thesameholdsforsparsematrices,forexamplebandmatrices.Ontheothersidethemethodshouldconverge,andthatasquicklyaspossible.Intheform(5.6)ΦisLipschitzcontinuousforagivennorm·onRmwithLipschitzconstantM,where·isanormonRm,mthatisconsistentwiththevectornorm(see(A3.9)).Moreprecisely,foraconsistentiterationtheerrore(k):=x(k)−x,withx=A−1bstilldenotingtheexactsolution,evensatisfiese(k+1)=Me(k),because(5.7)and(5.8)imply(k+1)(k+1)(k)(k)e=x−x=Mx+Nb−Mx−NAx=Me.(5.11)ThespectralradiusofM,thatis,themaximumoftheabsolutevaluesofthe(complex)eigenvaluesofM,willbedenotedby(M).Thefollowinggeneralconvergencetheoremholds:Theorem5.1Afixed-pointiterationgivenby(5.6)tosolveAx=bisgloballyandlinearlyconvergentif(M)<1.(5.12)Thisissatisfiedifforamatrixnorm·onRm,minducedbyanorm·onRmwehaveM<1.(5.13) 2025.IterativeMethodsforSystemsofLinearEquationsIftheconsistencycondition(5.8)holdsandthematrixandvectornormsappliedareconsistent,thentheconvergenceismonotoneinthefollowingsense:e(k+1)≤Me(k).(5.14)Proof:Assuming(5.12),thenforε=(1−(M))/2>0thereisanorm·onRmsuchthattheinducednorm·onRm,msatisfiesSSMS≤(M)+ε<1(see[16,p.34]).ThefunctionΦisacontractionwithrespecttothisspecialnormonRm.Therefore,Banach’sfixed-pointtheorem(Theorem8.4)canbeappliedonX=(Rm,·),whichensurestheglobalconvergenceofSthesequencex(k)toafixedpoint¯xofΦ.kIf(5.13)holds,Φisacontractionevenwithrespecttothenorm·onRm,andMistheLipschitzconstant.Finallyrelation(5.14)followsfrom(5.11).
Inanycase,wehaveconvergenceinanynormonRm,sincetheyareallequivalent.Linearconvergencefor(5.12)holdsonlyinthegenerallynotavailablenorm·SwithMSascontractionnumber.Asterminationcriterionfortheconcreteiterationmethodstobeintroduced,ofteng(k)≤δg(0)(5.15)isusedwithacontrolparameterδ>0,abbreviatedasg(k)=0.Theconnectiontothedesiredreductionoftherelativeerroraccordingto(5.2)isgivenbye(k)g(k)≤κ(A),(5.16)e(0)g(0)wheretheconditionnumberκ(A)=AA−1istobecomputedwithrespecttoamatrixnormthatisconsistentwiththechosenvectornorm.Relation(5.16)followsfrome(k)=A−1g(k)≤A−1g(k),g(0)=Ae(0)≤Ae(0).Therefore,fortheselectionofδin(5.15)wehavetotakeintoaccountthebehaviouroftheconditionnumber.FortheiterationmatrixM,accordingto(5.8),wehaveM=I−NA,oraccordingto(5.10)withnonsingularW,−1M=I−WA. 5.1.LinearStationaryIterativeMethods203Toimprovetheconvergence,i.e.toreduce(M)(orM),weneedN≈A−1andW≈A,whichisincontradictiontothefastsolvabilityof(5.10).5.1.2ClassicalMethodsThefastsolvabilityof(5.10)(inO(m)operations)isensuredbychoosingW:=D,(5.17)whereA=L+D+RistheuniquepartitionofA,withastrictlylowertriangularmatrixL,astrictlyuppertriangularmatrixR,andthediagonalmatrixD:0······00a1,2···a1,ma2,10···0........L:=......,R:=....,......0···0am−1,mam,1···am,m−100······0a11a220D:=...0.amm(5.18)Assumeaii=0foralli=1,...,m,orequivalentlythatDisnonsingular,whichcanbeachievedbyrowandcolumnpermutation.Thechoiceof(5.17)iscalledthemethodofsimultaneousdisplacementsorJacobi’smethod.Intheformulationform(5.6)wehaveN=D−1,M=I−NA=I−D−1A=−D−1(L+R).JTherefore,theiterationcanbewrittenas(k+1)(k)(k)Dx−x=−Ax−borx(k+1)=D−1−Lx(k)−Rx(k)+b(5.19)or1i−1mx(k+1)=−ax(k)−ax(k)+bforalli=1,...,m.iaijjijjiiij=1j=i+1Ontherightsideinthefirstsumitisreasonabletousethenewiteratex(k+1)whereitisalreadycalculated.Thisleadsustotheiteration(k+1)−1(k+1)(k)x=D−Lx−Rx+b(5.20) 2045.IterativeMethodsforSystemsofLinearEquationsor(k+1)(k)(D+L)x=−Rx+bor(D+L)x(k+1)−x(k)=−Ax(k)−b,(5.21)theso-calledmethodofsuccessivedisplacementsorGauss–Seidelmethod.Accordingto(5.21)wehavehereaconsistentiterationwithW=D+L.SinceDisnonsingular,Wisnonsingular.Writtenintheform(5.6)themethodisdefinedbyN=W−1=(D+L)−1,−1−1MGS=I−NA=I−(D+L)A=−(D+L)R.IncontrasttotheJacobiiteration,theGauss–Seideliterationdependsontheorderoftheequations.However,thederivation(5.20)showsthatthenumberofoperationsperiterationstepisequal,JacobibecomesGauss–Seidel,ifx(k+1)isstoredonthesamevectorasx(k).Asufficientconvergenceconditionisgivenbythefollowingtheorem:Theorem5.2Jacobi’smethodandtheGauss–Seidelmethodconvergegloballyandmonotonicallywithrespectto·∞ifthestrictrowsumcriterionm|aij|<|aii|foralli=1,...,m(5.22)j=1j=iissatisfied.Proof:TheproofhereisgivenonlyfortheJacobiiteration.Fortheothermethodsee,forexample,[16].Theinequality(5.22)isequivalenttoMJ∞<1becauseofMJ=−D−1(L+R)if·isthematrixnormthatisinducedby·,which∞∞meansthemaximum-row-sumnorm(see(A3.6)).
ItcanbeshownthattheGauss–Seidelmethodconverges“better”thanJacobi’smethod,asexpected:Undertheassumptionof(5.22)fortherespectiveiterationmatrices,MGS∞≤MJ∞<1(see,forexample,[16]). 5.1.LinearStationaryIterativeMethods205Theorem5.3IfAissymmetricandpositivedefinite,thentheGauss–Seidelmethodconvergesglobally.TheconvergenceismonotoneintheT1/2menergynorm·A,wherexA:=xAxforx∈R.Proof:See[16,p.90].
Ifthedifferentialoperator,andthereforethebilinearform,issymmet-ric,thatis,if(3.12)holdswithc=0,thenTheorem5.3canbeapplied.ConcerningtheapplicabilityofTheorem5.2,evenforthePoissonequationwithDirichletboundaryconditions(1.1),(1.2)requirementsforthefiniteelementdiscretizationarenecessaryinordertosatisfyatleastaweakerversionof(5.22).Thisexamplethensatisfiestheweakrowsumcriteriononlyinthefollowingsense:m|aij|≤|aii|foralli=1,...,m;j=1(5.23)j=i“<”holdsforatleastonei∈{1,...,m}.Inthecaseofthefinitedifferencemethod(1.7)fortherectangulardo-mainorthefiniteelementmethodfromSection2.2,whichleadstothesamediscretizationmatrix,(5.23)issatisfied.Forageneraltriangulationwithlinearansatzfunctions,conditionsfortheanglesoftheelementsmustberequired(seetheangleconditioninSection3.9).Thecondition(5.23)isalsosufficient,ifAisirreducible(seeAppendixA.3).Theorem5.4IfAsatisfiesthecondition(5.23)andisirreducible,thenJacobi’smethodconvergesglobally.Proof:See[28,p.111].
ThequalitativestatementofconvergencedoesnotsayanythingabouttheusefulnessofJacobi’sandtheGauss–Seidelmethodforfiniteelementdiscretizations.AsanexampleweconsidertheDirichletproblemforthePoissonequationonarectangulardomainasin(1.5),withthefive-pointstencildiscretizationintroducedinSection1.2.Werestrictourselvestoanequalnumberofnodesinbothspacedirectionsforsimplificityofthenotation.Thisnumberisdenotedbyn+1,differentlythaninChapter1.Therefore,A∈Rm,maccordingto(1.14),withm=(n−1)2beingthenumberofinteriornodes.Thefactorh−2canbeomittedbymultiplyingtheequationbyh2.Intheaboveexampletheeigenvaluesandthereforethespectralradiuscanbecalculatedexplicitly.DuetoD=4IwehaveforJacobi’smethod11M=−(A−4I)=I−A,44 2065.IterativeMethodsforSystemsofLinearEquationsandthereforeAandMhavethesameeigenvectors,namely,ikπjlπzk,l=sinsin,1≤i,j,k,l≤n−1,ijnnwiththeeigenvalueskπlπ22−cos−cos(5.24)nnforAand1kπ1lπcos+cos(5.25)2n2nforMwith1≤k,l≤n−1.Thiscanbeprovendirectlywiththehelpoftrigonometricidentities(see,forexample,[15,p.53]).Thuswehave(n−1)πππ2(M)=−cos=cos=1−+On−4.(5.26)nn2n2Withgrowingntherateofconvergencebecomesworse.Theefforttogainanapproximativesolution,whichmeanstoreducetheerrorlevelbelowagiventhresholdε,isproportionaltothenumberofiterations×operationsforaniteration,aswediscussedatthebeginningofthischapter.Dueto(5.4)and(5.12)thenumberofnecessaryoperationsiscalculatedasfollows:ln(1/ε)1212·O(m)=ln·On·O(m)=lnO(m).−ln((M))εεHerethewell-knownexpansionln(1+x)=x+O(x2)isemployedinthedeterminationoftheleadingtermof−1/(ln((M)).AnanalogousresultwithbetterconstantsholdsfortheGauss–Seidelmethod.Incomparisontothis,theeliminationortheCholeskymethodrequiresOband-width2·m=O(m2)operations;i.e.,theyareofthesamecomplexity.Therefore,bothmethodsareofuseforonlymoderatelylargem.AniterativemethodhasasuperiorcomplexitytothatoftheCholeskymethodif(M)=1−O(n−α)(5.27)withα<2.Intheidealcase(5.5)holds;thenthemethodneedsO(m)operations,whichisasymptoticallyoptimal.Inthefollowingwewillpresentasequenceofmethodswithincreasinglybetterconvergencepropertiesforsystemsofequationsthatarisefromfiniteelementdiscretizations.ThesimplestiterationistheRichardsonmethod,definedbyM=I−A,i.e.,N=W=I.(5.28) 5.1.LinearStationaryIterativeMethods207Forthismethodwehave(M)=max{|1−λmax(A)|,|1−λmin(A)|},withλmax(A)andλmin(A)beingthelargestandsmallesteigenvaluesofA,respectively.Therefore,thismethodisconvergentforspecialmatricesonly.InthecaseofanonsingularD,theRichardsonmethodforthetransformedsystemofequationsD−1Ax=D−1bisequivalenttoJacobi’smethod.Moregenerally,thefollowingcanbeshown:IfaconsistentmethodisdefinedbyM,NwithI=M+NA,andNnonsingular,thenitisequivalenttotheRichardsonmethodappliedtoNAx=Nb.(5.29)TheRichardsonmethodfor(5.29)hastheformx(k+1)−x(k)=−N˜NAx(k)−NbwithN˜=I,whichmeanstheform(5.9),andviceversa.Equation(5.29)canalsobeinterpretedasapreconditioningofthesystemofequations(5.1),withtheaimtoreducethespectralconditionnumberκ(A)ofthesystemmatrix,sincethisisessentialfortheconvergencebe-haviour.Thiswillbefurtherspecifiedinthefollowingconsiderations(5.33),(5.73).Asalreadyseenintheaforementionedexamples,thematrixNAwillnotbeconstructedexplicitly,sinceNisingeneraldenselyoccupied,evenifN−1issparse.Theevaluationofy=NAxthereforemeanssolvingtheauxiliarysystemofequations−1Ny=Ax.Obviously,wehavethefollowing:Lemma5.5IfthematrixAissymmetricandpositivedefinite,thenfortheRichardsonmethodalleigenvaluesofMarerealandsmallerthan1.5.1.3RelaxationWecontinuetoassumethatAissymmetricandpositivedefinite.Therefore,divergenceoftheprocedurecanbecausedonlybynegativeeigenvaluesofI−Alessthanorequalto−1.Ingeneral,badornonconvergentiterativemethodscanbeimprovedintheirconvergencebehaviourbyrelaxationiftheymeetcertainconditions.Foraniterationmethod,givenintheform(5.6),(5.7),thecorrespondingrelaxationmethodwithrelaxationparameterω>0isdefinedby(k+1)(k)(k)x:=ωMx+Nb+(1−ω)x,(5.30) 2085.IterativeMethodsforSystemsofLinearEquationswhichmeansMω:=ωM+(1−ω)I,Nω:=ωN,(5.31)oriftheconditionofconsistencyM=I−NAholds,x(k+1)=ωx(k)−NAx(k)−b+(1−ω)x(k)=x(k)−ωNAx(k)−b.Letusassumefortheprocedure(5.6)thatalleigenvaluesofMarereal.Forthesmallestoneλminandthelargestoneλmaxweassumeλmin≤λmax<1;thisis,forexample,thecasefortheRichardsonmethod.ThenalsotheeigenvaluesofMωarereal,andweconcludethatλi(Mω)=ωλi(M)+1−ω=1−ω1−λi(M)iftheλi(B)aretheeigenvaluesofBinanarbitraryordering.Hence(Mω)=max|1−ω(1−λmin(M))|,|1−ω(1−λmax(M))|,sincef(λ):=1−ω(1−λ)isastraightlineforafixedω(withf(1)=1andf(0)=1−ω).fρ(Mω)-1+ω(1−λmin)11λmaxω1-ω(1−λmax)1λminλmaxλωω1λmin11-ω(1−λmin)ω2..f(λ)forω1<1andω2>1Figure5.1.Calculationof¯ω.Fortheoptimalω¯,i.e.,¯ωwith(Mω¯)=min(Mω),ω>0wethereforehave,ascanbeseenfromFigure5.1,1−ω¯(1−λmax(M))=−1+¯ω(1−λmin(M))2⇐⇒ω¯=.2−λmax(M)−λmin(M)Hence¯ω>0and(Mω¯)=1−ω¯(1−λmax(M))<1; 5.1.LinearStationaryIterativeMethods209consequently,themethodconvergeswithoptimalωevenincaseswhereitwouldnotconvergeforω=1.ButkeepinmindthatoneneedstheeigenvaluesofMtodetermine¯ω.Moreover,wehaveω<¯1⇔λmax(M)+λmin(M)<0.Ifλmin(M)=−λmax(M),thatis,¯ω=1,wewillachieveanimprovementbyrelaxation:(Mω¯)<(M).Thecaseofω<1iscalledunderrelaxation,whereasinthecaseofω>1wespeakofanovererrelaxation.Inparticular,fortheRichardsonmethodwiththeiterationmatrixM=I−A,duetoλmin(M)=1−λmax(A)andλmax(M)=1−λmin(A),theoptimal¯ωisgivenby2ω¯=.(5.32)λmin(A)+λmax(A)Henceλmax(A)−λmin(A)κ(A)−1(Mω¯)=1−ωλ¯min(A)==<1,(5.33)λmin(A)+λmax(A)κ(A)+1withthespectralconditionnumberofAλmax(A)κ(A):=λmin(A)(seeAppendixA.3).Forlargeκ(A)wehaveκ(A)−12(Mω¯)=≈1−,κ(A)+1κ(A)thevariableoftheproportionalitybeingκ(A).Fortheexampleofthefive-pointstencildiscretization,dueto(5.24),n−1πλmin(A)+λmax(A)=42−cosπ−cos=8,nnandthusdueto(5.32),1ω¯=.4HencetheiterationmatrixM=I−1AisidenticaltotheJacobiiteration:ω¯4WehaverediscoveredJacobi’smethod.Bymeansof(5.33)wecanestimatethecontractionnumber,sinceweknowfrom(5.24)that41−cosn−1π1+cosπ4n2κ(A)=n=n≈.(5.34)41−cosπ1−cosππ2nn 2105.IterativeMethodsforSystemsofLinearEquationsThisshowsthestringencyofTheorem3.45,andagainwecanconcludethatππ2(Mω¯)=cos≈1−.(5.35)n2n2DuetoTheorem3.45theconvergencebehaviourseenforthemodelproblemisalsovalidingeneralforquasi-uniformtriangulations.5.1.4SORandBlock-IterationMethodsWeassumeagainthatAisageneralnonsingularmatrix.FortherelaxationoftheGauss–SeidelmethodweuseitintheformDx(k+1)=−Lx(k+1)−Rx(k)+b,insteadoftheresolvedform(5.20).Therelaxedmethodisthen(k+1)(k+1)(k)(k)Dx=ω−Lx−Rx+b+(1−ω)Dx(5.36)witharelaxationparameterω>0.Thisisequivalentto(k+1)(k)(D+ωL)x=(−ωR+(1−ω)D)x+ωb.(5.37)Hence−1Mω:=(D+ωL)(−ωR+(1−ω)D),−1Nω:=(D+ωL)ω.Intheapplicationtodiscretizationsofboundaryvalueproblems,nor-mallywechooseω>1,whichmeansoverrelaxation.ThisexplainsthenameoftheSORmethodasanabbreviationofsuccessiveoverrelaxation.TheefforttoexecuteaniterationstepishardlyhigherthanfortheGauss–Seidelmethod.Althoughwehavetoadd3moperationstotheevaluationoftherightsideof(5.36),theforwardsubstitutiontosolvetheauxiliarysystemofequationsin(5.37)isalreadypartoftheform(5.36).Thecalculationoftheoptimal¯ωhereismoredifficult,becauseMωdependsnonlinearlyonω.Onlyforspecialclassesofmatricescantheopti-mal¯ωminimizing(Mω)becalculatedexplicitlyindependenceon(M1),theconvergencerateofthe(nonrelaxed)Gauss–Seidelmethod.Beforewesketchthis,wewilllookatsomefurthervariantsofthisprocedure:ThematrixNωisnonsymmetricevenforsymmetricA.Onegetsasym-metricNωifafteroneSORstepanotheroneisperformedinwhichtheindicesarerunthroughinreverseorderm,m−1,...,2,1,whichmeansthatLandRareexchanged.Thetwohalfsteps11(k+)(k+)(k)(k)Dx2=ω−Lx2−Rx+b+(1−ω)Dx,11(k+1)(k+)(k+1)(k+)Dx=ω−Lx2−Rx+b+(1−ω)Dx2, 5.1.LinearStationaryIterativeMethods211makeuponestepofthesymmetricSOR,theSSORmethodforshort.AspecialcaseisthesymmetricGauss–Seidelmethodforω=1.WewritedowntheprocedureforsymmetricA,i.e.,R=LTintheform(5.6),inwhichthesymmetryofNbecomesobvious:T−1&'−1&T'M=D+ωL(1−ω)D−ωL(D+ωL)(1−ω)D−ωL,T−1−1N=ω(2−ω)D+ωLD(D+ωL).(5.38)TheeffortforSSORisonlyslightlyhigherthanforSORifthevectorsalreadycalculatedinthehalfstepsarestoredandusedagain,asforexampleLx(k+1/2).OthervariantsoftheseproceduresarecreatediftheproceduresarenotappliedtothematrixitselfbuttoablockpartitioningA=(A)withA∈Rmi,mj,i,j=1,...,p,(5.39)iji,jijpwithi=1mi=m.Asanexamplewegettheblock-Jacobimethod,whichisanalogousto(5.19)andhastheformi−1pξ(k+1)=A−1−Aξ(k)−Aξ(k)+βforalli=1,...,p.iiiijjijjij=1j=i+1(5.40)Herex=(ξ,...,ξ)Tandb=(β,...,β)T,respectively,arecorrespond-1p1p(k)(k+1)ingpartitionsofthevectors.Byexchangingξjwithξjinthefirstsumonegetstheblock-Gauss–Seidelmethodandtheninthesamewaytherelaxedvariants.Theiteration(5.40)includespvectorequations.ForeachofthemwehavetosolveasystemofequationswithsystemmatrixAii.Togetanadvantagecomparedtothepointwisemethodamuchlowereffortshouldbenecessarythanforthesolutionofthetotalsystem.Thiscanrequire—ifatallpossible—arearrangingofthevariablesandequa-tions.Thenecessarypermutationswillnotbenotedexplicitlyhere.Suchmethodsareappliedinfinitedifferencemethodsorothermethodswithstructuredgrids(seeSection4.1)ifanorderingofnodesispossiblesuchthatthematricesAiiarediagonalortridiagonalandthereforethesystemsofequationsaresolvablewithO(mi)operations.Asanexampleweagaindiscussthefive-pointstencildiscretizationofthePoissonequationonasquarewithn+1nodesperspacedimension.ThematrixAthenhastheform(1.14)withl=m=n.Ifthenodesarenumberedrowwiseandwechooseoneblockforeachline,whichmeansp=n−1andmi=n−1foralli=1,...,p,thenthematricesAiiaretridiagonal.Ontheotherhand,ifonechoosesapartitionoftheindicesofthenodesinsubsetsSisuchthatanodewithindexinSihasneighboursonlyinotherindexsets,thenforsuchaselectionandarbitraryorderingwithintheindexsetsthematricesAiibecomediagonal.Neighboursheredenotethenodeswithinadifferencestencilormoregenerally,thosenodes 2125.IterativeMethodsforSystemsofLinearEquations4−10−100000−14−10−100000−1400−1000−1004−10−1000−10−14−10−1000−10−1400−1000−1004−100000−10−14−100000−10−14m=3×3:rowwiseordering.40000−1−10004000−10−1000400−1−1−1−1000400−10−10000400−1−1−1−1−1004000−10−1−1004000−1−10−1004000−1−1−10004red-blackordering:red:node1,3,5,7,9fromrowwiseorderingblack:node2,4,6,8fromrowwiseorderingFigure5.2.Comparisonoforderings.thatcontributetothecorrespondingrowofthediscretizationmatrix.Intheexampleofthefive-pointstencil,startingwithrowwisenumbering,onecancombinealloddindicestoablockS1(the“rednodes”)andallevenindicestoablockS2(the“black”nodes).Herewehavep=2.Wecallthisared-blackordering(seeFigure5.2).Iftwo“colours”arenotsufficient,onecanchoosep>2.WereturntotheSORmethodanditsconvergence:InthefollowingtheiterationmatrixwillbedenotedbyMSOR(ω)withtherelaxationparameterω.Likewise,MJandMGSaretheiterationmatricesofJacobi’sandtheGauss–Seidelmethod,respectively.Generalpropositionsaresummarizedinthefollowingtheorem: 5.1.LinearStationaryIterativeMethods213Theorem5.6(ofKahan;OstrowskiandReich)(1)MSOR(ω)≥|1−ω|forω=0.(2)IfAissymmetricandpositivedefinite,thenMSOR(ω)<1forω∈(0,2).Proof:See[16,pp.91f.].
Therefore,weuseonlyω∈(0,2).Forausefulprocedureweneedmoreinformationabouttheoptimalrelaxationparameterωopt,givenbyMSOR(ωopt)=minMSOR(ω),0<ω<2andaboutthesizeofthecontractionnumber.Thisispossibleonlyiftheorderingofequationsandunknownshascertainproperties:Definition5.7AmatrixA∈Rm,misconsistentlyorderedifforthepartition(5.18),Disnonsingularand−1−1−1C(α):=αDL+αDRhaseigenvaluesindependentofαforα∈C{0}.Thereisaconnectiontothepossibilityofamulti-colourordering,becauseamatrixintheblockform(5.39)isconsistentlyorderedifitisblock-tridiagonal(i.e.,Aij=0for|i−j|>1)andthediagonalblocksAiiarenonsingulardiagonalmatrices(see[28,pp.114f.]).InthecaseofaconsistentlyorderedmatrixonecanprovearelationbetweentheeigenvaluesofMJ,MGS,andMSOR(ω).FromthiswecanseehowmuchfastertheGauss–SeidelmethodconvergesthanJacobi’smethod:Theorem5.8IfAisconsistentlyordered,then(M)2=(M).JGSProof:ForaspecialcaseseeRemark5.5.2in[16].
Dueto(5.4)wecanexpectahalvingofthenumberofiterationsteps,butthisdoesnotchangetheasymptoticstatement(5.27).Finally,inthecasethatJacobi’smethodconvergesthefollowingtheoremholds:Theorem5.9LetAbeconsistentlyorderedwithnonsingulardiagonalma-trixD,theeigenvaluesofMJbeingrealandβ:=(MJ)<1.ThenwehavefortheSORmethod:2(1)ωopt=21/2,1+(1−β) 2145.IterativeMethodsforSystemsofLinearEquations221/21−ω+1ω2β2+ωβ1−ω+ωβ24(2)(MSOR(ω))=for0<ω<ωoptω−1forωopt≤ω<2,β2(3)MSOR(ωopt)=21/22.(1+(1−β))Proof:See[18,p.216].
ρ(MSOR(ω))1012ωoptωFigure5.3.DependenceofMSOR(ω)onω.If(MJ)isknownforJacobi’smethod,thenωoptcanbecalculated.Thisisthecaseintheexampleofthefive-pointstencildiscretizationonasquare:From(5.26)andTheorem5.9itfollowsthatπ2π2−4(MGS)=cos=1−+O(n);nn2henceπωopt=2/1+sinn,MSOR(ωopt)=ωopt−1=1−2πn+O(n−2).Therefore,theoptimalSORmethodhasalowercomplexitythanallmethodsdescribeduptonow.Correspondingly,thenumberofoperationstoreachtherelativeer-rorlevelε>0isreducedtoln1O(m3/2)operationsincomparisontoεln1O(m2)operationsforthepreviousprocedures.εTable5.1givesanimpressionoftheconvergenceforthemodelproblem.Itdisplaysthetheoreticallytobeexpectedvaluesforthenumbersofiter-ationsoftheGauss–Seidelmethod(mGS),aswellasfortheSORmethod 5.1.LinearStationaryIterativeMethods215nmGSmSOR843816178173271535642865701281146614025645867281Table5.1.Gauss–SeidelandoptimalSORmethodforthemodelproblem.withoptimalrelaxationparameter(mSOR).Hereweusetheverymoderateterminationcriterionε=10−3measuredintheEuclideannorm.TheoptimalSORmethodissuperior,evenifwetakeintoaccountthealmostdoubledeffortperiterationstep.Butgenerally,ωoptisnotknownexplicitly.Figure5.3showsthatitisprobablybettertooverestimateωoptinsteadofunderestimating.Moregenerally,onecantrytoimprovetherelaxationparameterduringtheiteration:If(MJ)isasimpleeigenvalue,thenthisalsoholdstrueforthespectralradius(MSOR(ω)).Thespectralradiuscanthusbeapproximatedbythepowermethodonthebasisoftheiterates.ByTheorem5.9(3)onecanapproximate(MJ),andbyTheorem5.9(1)thenalsoωopt.Thisbasicprinciplecanbeextendedtoanalgorithm(see,forexample,[18,Section9.5]),buttheupcomingoverallprocedureisnolongeralinearstationarymethod.5.1.5ExtrapolationMethodsAnotherpossibilityforanextensionofthelinearstationarymethods,re-latedtotheadaptionoftherelaxationparameter,isthefollowing:Startingwithalinearstationarybasiciteration˜xk+1:=Φx˜kwedefineanewiterationbyx(k+1):=ωΦx(k)+(1−ω)x(k),(5.41)kkwithextrapolationfactorsωktobechosen.Ageneralizationofthisdefi-nitionistostartwiththeiteratesofthebasiciteration˜x(0),x˜(1),....Theiteratesofthenewmethodaretobedeterminedbykx(k):=αx˜(j),kjj=0withαkjdefinedbyapolynomialpk∈Pk,withthepropertypk(t)=kjj=0αkjtandpk(1)=1.Foranappropriatedefinitionofsuchextrapola-tionorsemi-iterativemethodsweneedtoknowthespectrumofthebasiciterationmatrixM,sincetheerrore(k)=x(k)−xsatisfies(k)(0)e=pk(M)e, 2165.IterativeMethodsforSystemsofLinearEquationswhereMistheiterationmatrixofthebasiciteration.Thismatrixshouldbenormal,forexample,suchthatpk(M)2=(pk(M))holds.Thenwehavetheobviousestimatione(k)≤p(M)e(0)≤p(M)e(0)≤(p(M))e(0).(5.42)2k2k22k2Ifthemethodistobedefinedinsuchawaythat(pk(M))=max|pk(λ)|λ∈σ(M)isminimizedbychoosingpk,thentheknowledgeofthespectrumσ(M)isnecessary.Generally,insteadofthis,weassumethatsuitablesupersetsareknown:Ifσ(M)isrealanda≤λ≤bforallλ∈σ(M),then,duetoe(k)≤maxp(λ)e(0),2k2λ∈[a,b]itmakessensetodeterminethepolynomialspkasasolutionoftheminimizationproblemon[a,b],max|pk(λ)|→minforallp∈Pkwithp(1)=1.(5.43)λ∈[a,b]Inthefollowingsectionswewillintroducemethodswithananalogousconvergencebehaviour,withoutcontrolparametersnecessaryfortheirdefinition.Forfurtherinformationonsemi-iterativemethodssee,forexample,[16,Chapter7].Exercises5.1InvestigateJacobi’smethodandtheGauss–SeidelmethodforsolvingthelinearsystemofequationsAx=bwithrespecttotheirconvergenceifwehavethefollowingsystemmatrices:12−22−111(a)A=111,(b)A=222.2221−1−125.2ProvetheconsistencyoftheSORmethod.5.3ProveTheorem5.6,(1). 5.2.GradientandConjugateGradientMethods2175.2GradientandConjugateGradientMethodsInthissectionletA∈Rm,mbesymmetricandpositivedefinite.ThenthesystemofequationsAx=bisequivalenttotheproblem1TTmMinimizef(x):=xAx−bxforx∈R,(5.44)2sinceforsuchafunctionaltheminimaandstationarypointscoincide,whereastationarypointisanxsatisfying0=∇f(x)=Ax−b.(5.45)Incontrasttothenotationx·yforthe“short”spacevectorsx,y∈RdwewriteheretheEuclideanscalarproductasmatrixproductxTy.ForthefiniteelementdiscretizationthiscorrespondstotheequivalenceoftheGalerkinmethod(2.23)withtheRitzmethod(2.24)ifAisthestiffnessmatrixandbtheloadvector(see(2.34)and(2.35)).Moregenerally,Lemma2.3impliestheequivalenceof(5.44)and(5.45),ifasbilinearformtheso-calledenergyscalarproductx,y:=xTAy(5.46)Aischosen.Ageneraliterativemethodtosolve(5.44)hasthefollowingstructure:Defineasearchdirectiond(k).Minimizeα→f˜(α):=fx(k)+αd(k)(5.47)exactlyorapproximately,withthesolutionαk.Definex(k+1):=x(k)+αd(k).(5.48)kIffisdefinedasin(5.44),theexactαkcanbecomputedfromtheconditionf˜
(α)=0andTf˜
(α)=∇fx(k)+αd(k)d(k)as(k)T(k)gdαk=−,(5.49)d(k)TAd(k)where(k)(k)(k)g:=Ax−b=∇fx.(5.50)Theerrorofthekthiterateisdenotedbye(k):e(k):=x(k)−x.Somerelationsthatarevalidinthisgeneralfromeworkarethefollowing:Duetotheone-dimensionalminimizationoff,wehave(k+1)T(k)gd=0,(5.51) 2185.IterativeMethodsforSystemsofLinearEquationsandfrom(5.50)wecanconcludeimmediatelythatAe(k)=g(k),e(k+1)=e(k)+αd(k),(5.52)kg(k+1)=g(k)+αAd(k).(5.53)kWeconsidertheenergynormT1/2xA:=xAx(5.54)inducedbytheenergyscalarproduct.ForafiniteelementstiffnessmatrixAwithabilinearformawehavethecorrespondence1/2xA=a(u,u)=uamforu=i=1xiϕiiftheϕiaretheunderlyingbasisfunctions.Comparingthesolutionx=A−1bwithanarbitraryy∈Rmleadsto12f(y)=f(x)+y−xA,(5.55)2sothatcondition(5.44)alsominimizesthedistancetoxin·A.Theenergynormwillthereforehaveaspecialimportance.Measuredintheenergynormwehave,dueto(5.52),e(k)2(k)T(k)(k)T−1(k)=eg=gAg,Aandthereforedueto(5.52)and(5.51),e(k+1)2(k+1)T(k)=ge.AThevector−∇fx(k)inx(k)pointsinthedirectionofthelocallysteepestdescent,whichmotivatesthegradientmethod,i.e.,d(k):=−g(k),(5.56)andthusd(k)Td(k)αk=.(5.57)d(k)TAd(k)Theaboveidentitiesimplyforthegradientmethod(k)T(k)e(k+1)2=g(k)+αAd(k)Te(k)=e(k)21−αddkAkTd(k)A−1d(k)andthusbymeansofthedefinitionofαkfrom(5.57)2d(k)Td(k)x(k+1)2x(k)2−x=−x1−.AAd(k)TAd(k)d(k)TA−1d(k)WiththeinequalityofKantorovich(see,forexample,[28,p.132]),TT−12xAxxAx11/21−1/2≤κ+κ,(xTx)222 5.2.GradientandConjugateGradientMethods219whereκ:=κ(A)isthespectralconditionnumber,andtherelation24(a−1)1−2=2fora>0,a1/2+a−1/2(a+1)weobtainthefollowingtheorem:Theorem5.10Forthegradientmethodwehavekκ−1x(k)−x≤x(0)−x.(5.58)Aκ+1AThisisthesameestimateasfortheoptimallyrelaxedRichardsonmethod(withthesharperestimateM≤κ−1insteadof(M)≤κ−1).TheAκ+1κ+1essentialdifferenceliesinthefactthatthisispossiblewithoutknowledgeofthespectrumofA.Nevertheless,forfiniteelementdiscretizationswehavethesamepoorconvergencerateasforJacobi’sorsimilarmethods.Thereasonforthisdeficiencyliesinthefactthatdueto(5.51),wehaveg(k+1)Tg(k)=0,butingeneralnotg(k+2)Tg(k)=0.Onthecontrary,thesesearchdirectionsareveryoftenalmostparallel,ascanbeseenfromFigure5.4.m=2:.(0)xf=constant(contourlines)Figure5.4.Zigzagbehaviourofthegradientmethod.Thereasonforthisproblemisthefactthatforlargeκthesearchdi-rectionsg(k)andg(k+1)canbealmostparallelwithrespecttothescalarproducts·,·A(seeExercise5.4),butwithrespectto·Athedistancetothesolutionwillbeminimized(see(5.55)).Thesearchdirectionsd(k)shouldbeorthogonalwithrespectto·,·,Awhichwecallconjugate.Definition5.11Vectorsd(0),...,d(l)∈Rmareconjugateiftheysatisfy34d(i),d(j)=0fori,j=0,...,l,i=j.AIfthesearchdirectionsofamethoddefinedaccordingto(5.48),(5.49)arechosenasconjugate,itiscalledamethodofconjugatedirections.Letd(0),...,d(m−1)beconjugatedirections.Thentheyarealsolinearlyindependentandthusformabasisinwhichthesolutionxof(5.1)canbe 2205.IterativeMethodsforSystemsofLinearEquationsrepresented,saybythecoefficientsγk:m−1x=γd(k).kk=0Sincethed(k)areconjugateandAx=bholds,wehave(k)Tdbγk=,(5.59)d(k)TAd(k)andtheγcanbecalculatedwithoutknowledgeofx.Ifthed(k)wouldbykgivenapriori,forexamplebyorthogonalizationofabasiswithrespectto·,·,thenxwouldbedeterminedby(5.59).AIfweapply(5.59)todeterminethecoefficientsforx−x(0)intheformm−1x−x(0)=γd(k),kk=0whichmeansreplacingbwithb−Ax(0)in(5.59),thenweget(0)T(k)gdγk=−.d(k)TAd(k)Forthekthiteratewehave,accordingto(5.48);k−1x(k)=x(0)+αd(i)ii=0andtherefore(see(5.50))k−1(k)(0)(i)g=g+αiAd.i=0Foramethodofconjugatedirectionsthisimplies(k)T(k)(0)T(k)gd=gdandthereforeg(k)Td(k)γk=−T=αk,d(k)Ad(k)whichmeansthatx=x(m).Amethodofconjugatedirectionsthereforeisexactafteratmostmsteps.Undercertainconditionssuchamethodmayterminatebeforereachingthisstepnumberwithg(k)=0andthefinaliteratex(k)=x.Ifmisverylarge,thisexactnessofamethodofconjugatedirectionsislessimportantthanthefactthattheiteratescanbeinterpretedasthesolutionofaminimizationproblemapproximating(5.44): 5.2.GradientandConjugateGradientMethods221Theorem5.12Theiteratesx(k)thataredeterminedbyamethodofcon-jugatedirectionsminimizethefunctionalffrom(5.44)aswellastheerrorx(k)−xonx(0)+K(A;g(0)),whereAkK(A;g(0)):=spand(0),...,d(k−1).kThisisdueto(k)T(i)gd=0fori=0,...,k−1.(5.60)Proof:Itissufficenttoprove(5.60).Duetotheone-dimensionalmini-mizationthisholdsfork=1andfori=k−1(see(5.51)appliedtok−1).Toconcludetheassertionforkfromitsknowledgefork−1,wenotethat(5.53)implies,for0≤iiA34canbeobtained.Thenecessaryrequirementd(k+1),d(k)=0leadstoA3434−g(k+1),d(k)+βd(k),d(k)=0⇐⇒AkA(k+1)T(k)gAdβk=T.(5.62)d(k)Ad(k)Inapplyingthemethoditisrecommendednottocalculateg(k+1)directlybuttouse(5.53)instead,becauseAd(k)isalreadynecessarytodetermineαkandβk.Thefollowingequivalenceshold:Theorem5.13IncasetheCGmethoddoesnotterminateprematurelywithx(k−1)beingthesolutionof(5.1),thenwehavefor1≤k≤mK(A;g(0))=spang(0),Ag(0),...,Ak−1g(0)k(5.63)=spang(0),...,g(k−1).Furthermore,g(k)Tg(i)=0fori=0,...,k−1,and(5.64)dimK(A;g(0))=k.k 2225.IterativeMethodsforSystemsofLinearEquationsThespaceK(A;g(0))=spang(0),Ag(0),...,Ak−1g(0)iscalledthekKrylov(sub)spaceofdimensionkofAwithrespecttog(0).Proof:Theidentities(5.64)areimmediateconsequencesof(5.63)andTheorem5.12.Theproofof(5.63)isgivenbyinduction:Fork=1theassertionistrivial.Letusassumethatfork≥1theidentity(5.63)holdsandthereforealso(5.64)does.Dueto(5.53)(appliedtok−1)itfollowsthat&'g(k)∈AKA;g(0)⊂spang(0),...,Akg(0)kandthusspang(0),...,g(k)=spang(0),...,Akg(0),becausetheleftspaceiscontainedintherightoneandthedimensionoftheleftsubspaceismaximal(=k+1)dueto(5.64)andg(i)=0foralli=0,...,k.Theidentityspand(0),...,d(k)=spang(0),...,g(k)followsfromtheinductionhypothesisand(5.61).
Thenumberofoperationsperiterationcanbereducedtoonematrixvector,twoscalarproducts,andthreeSAXPYoperations,ifthefollowingequivalenttermsareused:g(k)Tg(k)g(k+1)Tg(k+1)αk=T,βk=T.(5.65)d(k)Ad(k)g(k)g(k)HereaSAXPYoperationisoftheformz:=x+αyforvectorsx,y,zandascalarα.Theidentities(5.65)canbeseenasfollows:Concerningαkwenotethatbecauseof(5.51)and(5.61),TTT−g(k)d(k)=−g(k)−g(k)+βd(k−1)=g(k)g(k),k−1andconcerningβk,becauseof(5.53),(5.64),(5.62),andtheidentity(5.49)forαk,wehaveTTTTg(k+1)g(k+1)=g(k+1)g(k)+αAd(k)=αg(k+1)Ad(k)=βg(k)g(k)kkkandhencetheassumption.ThealgorithmissummarizedinTable5.2.Indeed,thealgorithmdefinesconjugatedirections:Theorem5.14Ifg(k−1)=0,thend(k−1)=0andthed(0),...,d(k−1)areconjugate. 5.2.GradientandConjugateGradientMethods223Chooseanyx(0)∈Rmandcalculated(0):=−g(0)=b−Ax(0).Fork=0,1,...putg(k)Tg(k)αk=,d(k)TAd(k)(k+1)(k)(k)x=x+αkd,g(k+1)=g(k)+αAd(k),k(k+1)T(k+1)ggβk=,g(k)Tg(k)(k+1)(k+1)(k)d=−g+βkd,untiltheterminationcriterion(“|g(k+1)|=0”)isfulfilled.2Table5.2.CGmethod.Proof:Theproofisdonebyinduction:Thecasek=1isclear.Assumethatd(0),...,d(k−1)areallnonzeroandconjugate.ThusaccordingtoTheorem5.12andTheorem5.13theidentities(5.60)–(5.64)holduptoindexk.Letusfirstprovethatd(k)=0:Duetog(k)+d(k)=βd(k−1)∈K(A;g(0))theassertiond(k)=0k−1kwouldimplydirectlyg(k)∈K(A;g(0)).Butrelations(5.63)and(5.64)kimplyfortheindexk,(k)T(0)gx=0forallx∈Kk(A;g),whichcontradictsg(k)=0.Inordertoproved(k)TAd(i)=0fori=0,...,k−1,accordingto(5.62)wehavetoproveonlythecasei≤k−2.Wehave(i)T(k)(i)T(k)(i)T(k−1)dAd=−dAg+βk−1dAd.ThefirsttermdisappearsduetoAd(i)∈AKA;g(0)⊂KA;g(0),k−1kwhichmeansthatAd(i)∈spand(0),...,d(k−1),and(5.60).Thesecondtermdisappearsbecauseoftheinductionhypothesis.
MethodsthataimatminimizingtheerrororresidualonKA;g(0)kwithrespecttoanorm·arecalledKrylovsubspacemethods.Heretheerrorwillbeminimizedintheenergynorm·=·Aaccordingto(5.55)andTheorem5.12.DuetotherepresentationoftheKrylovspaceinTheorem5.13theelementsy∈x(0)+KA;g(0)areexactlythevectorsoftheformky=x(0)+q(A)g(0),foranyq∈P(forthenotationq(A)seeAppendixk−1 2245.IterativeMethodsforSystemsofLinearEquationsA.3).Henceitfollowsthat(0)(0)(0)y−x=x−x+q(A)Ax−x=p(A)x−x,withp(z)=1+q(z)z,i.e.,p∈Pkandp(0)=1.Ontheotherhand,anysuchpolynomialcanberepresentedinthegivenform(defineqbyq(z)=(p(z)−1)/z).ThusTheorem5.12impliesx(k)−x≤y−x=p(A)x(0)−x(5.66)AAAforanyp∈Pkwithp(0)=1.Letz1,...,zmbeanorthonormalbasisofeigenvectors,thatis,Az=λzandzTz=δfori,j=1,...,m.(5.67)jjjijij(0)mThenwehavex−x=j=1cjzjforcertaincj∈R,andhencemp(A)x(0)−x=p(λ)czjjjj=1andthereforemmx(0)−x2=x(0)−xTAx(0)−x=cczTAz=λ|c|2Aijijjji,j=1j=1andanalogouslym2p(A)x(0)−x2=λ|cp(λ)|2≤max|p(λ)|x(0)−x2.AjjjiAi=1,...,mj=1(5.68)Relations(5.66),(5.68)implythefollowingtheorem:Theorem5.15FortheCGmethodandanyp∈Pksatisfyingp(0)=1,wehavex(k)−x≤max|p(λ)|x(0)−x,AiAi=1,...,mwiththeeigenvaluesλ1,...,λmofA.IftheeigenvaluesofAarenotknown,buttheirlocationis,i.e.,ifoneknowsa,b∈Rsuchthata≤λ1,...,λm≤b,(5.69)thenonlythefollowingweakerestimatecanbeused:x(k)−x≤max|p(λ)|x(0)−x.(5.70)AAλ∈[a,b]Therefore,wehavetofindp∈Pmwithp(0)=1thatminimizesmax|p(λ)|λ∈[a,b]. 5.2.GradientandConjugateGradientMethods225Thisapproximationprobleminthemaximumnormappearedalreadyin(5.43),becausethereisabijectionbetweenthesetsp∈Pkp(1)=1andp∈Pkp(0)=1throughp→p,˜p˜(t):=p(1−t).(5.71)ItssolutioncanrepresentedbyusingtheChebyshevpolynomialsofthefirstkind(see,forexample,[38,p.302]).TheyarerecursivelydefinedbyT0(x):=1,T1(x):=x,Tk+1(x):=2xTk(x)−Tk−1(x)forx∈RandhavetherepresentationTk(x)=cos(karccos(x))for|x|≤1.Thisimmediatelyimplies|Tk(x)|≤1for|x|≤1.Afurtherrepresentation,validforx∈R,iskk121/221/2Tk(x)=x+x−1+x−x−1.(5.72)2Theoptimalpolynomialin(5.70)isthendefinedbyTk((b+a−2z)/(b−a))p(z):=forz∈R.Tk((b+a)/(b−a))Thisimpliesthefollowingresult:Theorem5.16LetκbethespectralconditionnumberofAandassumeκ>1.Thenk1κ1/2−1x(k)−x≤x(0)−x≤2x(0)−x.(5.73)ATκ+1Aκ1/2+1Akκ−1Proof:Chooseaasthesmallesteigenvalueλminandbasthelargestoneλmax.Thefirstinequalityfollowsimmediatelyfrom(5.70)andκ=b/a.Forthesecondinequalitynotethatdueto(κ+1)/(κ−1)=1+2/(κ−1)=:1+2η≥1,(5.72)impliesκ+1121/2kTk≥1+2η+(1+2η)−1κ−121k1/2=1+2η+2(η(η+1)).2Finally,1/21/21/22(η+1)1/2+η1/21+2η+2(η(η+1))=η+(η+1)=(η+1)1/2−η1/2 2265.IterativeMethodsforSystemsofLinearEquations1/2(1+1/η)+1=,1/2(1+1/η)−1whichconcludestheproofbecauseof1+1/η=κ.
Forlargeκwehaveagainκ1/2−12≈1−.κ1/2+1κ1/2Comparedwith(5.58),κhasbeenimprovedtoκ1/2.From(5.4)and(5.34)thecomplexityofthefive-pointstencildiscretiza-tionofthePoissonequationonthesquareresultsin1lnOκ1/2O(m)=O(n)O(m)=Om3/2.εThisisthesamebehaviourasthatoftheSORmethodwithoptimalre-laxationparameter.TheadvantageoftheabovemethodliesinthefactthatthedeterminationofparametersisnotnecessaryforapplyingtheCGmethod.Forquasi-uniformtriangulations,Theorem3.45impliesananalogousgeneralstatement.Arelationtothesemi-iterativemethodsfollowsfrom(5.71):Theestimate(5.66)canalsobeexpressedase(k)≤p(I−A)e(0)(5.74)AAforanyp∈Pkwithp(1)=1.Thisisthesameestimateas(5.42)fortheRichardsoniteration(5.28)asbasismethod,withtheEuclideannorm|·|2replacedbytheenergynorm·A.Whilethesemi-iterativemethodsaredefinedbyminimizationofupperboundsin(5.42),theCGmethodisoptimalinthesenseof(5.74),withoutknowledgeofthespectrumσ(I−A).InthismannertheCGmethodcanbeseenasan(optimal)accelerationmethodfortheRichardsoniteration.Exercises5.4LetA∈Rm,mbeasymmetricpositivedefinitematrix.(a)Showthatforx,ywithxTy=0wehavex,yκ−1A≤,xAyAκ+1whereκdenotesthespectralconditionnumberofA.Hint:Representx,yintermsofanorthonormalbasisconsistingofeigenvectorsofA. 5.3.PreconditionedConjugateGradientMethod227(b)Showusingtheexamplem=2thatthisestimateissharp.Tothisend,lookforapositivedefinitesymmetricmatrixA∈R2,2aswellasvectorsx,y∈R2withxTy=0andx,yκ−1A=.xAyAκ+15.5ProvethatthecomputationoftheconjugatedirectionintheCGmethodinthegeneralstepk≥2isequivalenttothethree-termrecursionformula(k+1)(k)(k−1)d=[αkA+(βk+1)I]d−βk−1d.5.6LetA∈Rm,mbeasymmetricpositivedefinitematrixwithspectralconditionnumberκ.Supposethatthespectrumσ(A)ofthematrixAsatisfiesa0∈σ(A)aswellasσ(A){a0}⊂[a,b]with00forx=0.CChoosingtheCGmethodfor(5.75)withrespectto·,·C,weobtainpreciselytheabovemethod.Incasetheterminationcriterion“g(k+1)=0”isusedfortheiteration,2thescalarproductmustbeadditionallycalculated.Alternatively,wemayTuse“g(k+1)h(k+1)=0”.Thentheresidualismeasuredinthenorm·C−1.FollowingthereasoningattheendofSection5.2,thePCGmethodcanbeinterpretedasanaccelerationofalinearstationarymethodwithiterationmatrixM=I−C−1A.Foraconsistentmethod,wehaveN=C−1or,intheformulation(5.10),W=C.ThisobservationcanbeextendedinsuchawaythattheCGmethodcanbeusedfortheaccelerationofiterationmethods,forexamplealsoforthemultigridmethod,whichwillbeintroducedinSection5.5.Due 2305.IterativeMethodsforSystemsofLinearEquationstothedeductionofthepreconditionedCGmethodandtheidentityx(k)−x=x˜(k)−x˜,ABwhichresultsfromthetransformation(5.76),theapproximationpropertiesfortheCGmethodalsoholdforthePCGmethodifthespectralconditionnumberκ(A)isreplacedbyκ(B)=κ(C−1A).Therefore,kκ1/2−1x(k)−x≤2x(0)−xAκ1/2+1Awithκ=κ(C−1A).ThereisacloserelationbetweenthosepreconditioningmatricesC,whichkeepκ(C−1A)small,andwell-convergentlinearstationaryiterationmeth-odswithN=C−1(andM=I−C−1A)ifNissymmetricandpositivedefinite.Indeed,−1κ(CA)≤(1+(M))/(1−(M))ifthemethoddefinedbyMandNisconvergentandNissymmetricforsymmetricA(seeExercise5.7).Fromtheconsideredlinearstationarymethodsbecauseoftherequiredsymmetrywemaytake•Jacobi’smethod:Thiscorrespondsexactlytothediagonalscaling,whichmeansthedivisionofeachequationbyitsdiagonalelement.Indeed,fromthedecomposition(5.18)wehaveC=N−1=D,andthePCGmethodisequivalenttothepreconditioningfromtheleftbythematrixC−1incombinationwiththeusageoftheenergyscalarproduct·,·C.•TheSSORmethod:Accordingto(5.38)wehaveC=ω−1(2−ω)−1(D+ωL)D−1(D+ωLT).HenceCissymmetricandpositivedefinite.Thesolutionoftheauxiliarysystemsofequationsneedsonlyforwardandbackwardsubstitutionswiththesamestructureofthematrixasforthesystemmatrix,sothattherequirementoflowercomplexityisalsofulfilled.Anexactestimationofκ(C−1A)shows(see[3,pp.328ff.])thatundercertainrequirementsforA,whichreflectpropertiesoftheboundaryvalueproblemandthediscretiza-tion,wefindaconsiderableimprovementoftheconditioningbyusinganestimateofthetypeκ(C−1A)≤const(κ(A)1/2+1).Thechoiceoftherelaxationparameterωisnotcritical.Insteadoftry-ingtochooseanoptimaloneforthecontractionnumberoftheSSOR 5.3.PreconditionedConjugateGradientMethod231method,wecanminimizeanestimationforκ(C−1A)(see[3,p.337]),whichrecommendsachoiceofωin[1.2,1.6].Forthefive-pointstencildiscretizationofthePoissonequationonthesquarewehave,accordingto(5.34),κ(A)=O(n2),andtheabovecon-ditionsarefulfilled(see[3,pp.330f.]).BySSORpreconditioningthisisimprovedtoκ(C−1A)=O(n),andthereforethecomplexityofthemethodis11lnOκ1/2O(m)=lnOn1/2O(m)=Om5/4.(5.77)εεAsdiscussedinSection2.5,directeliminationmethodsarenotsuitableinconjunctionwiththediscretizationofboundaryvalueproblemswithlargenodenumbers,becauseingeneralfill-inoccurs.AsdiscussedinSection2.5,L=(lij)describesalowertriangularmatrixwithlii=1foralli=1,...,m(thedimensionisdescribedtherewiththenumberofdegreesoffreedomM)andU=(uij)anuppertriangularmatrix.TheideaoftheincompleteLUfactorization,orILUfactorization,istoallowonlycertainpatternsE∈{1,...,m}2fortheentriesofLandU,andinsteadofA=LU,ingeneralwecanrequireonlyA=LU−R.HeretheremainderR=(r)∈Rm,mhastosatisfyijrij=0for(i,j)∈E.(5.78)Therequirementsmaij=likukjfor(i,j)∈E(5.79)k=1mean|E|equationsforthe|E|entriesofthematricesLandU.(Noticethatlii=1foralli.)Theexistenceofsuchfactorizationswillbediscussedlater.AnalogouslytothecloseconnectionbetweentheLUfactorizationandanLDLTorLLTfactorizationforsymmetricorsymmetricpositivedef-initematrices,asdefinedinSection2.5,wecanusetheICfactorization(incompleteCholeskyfactorization)forsuchmatrices.TheICfactorizationneedsarepresentationinthefollowingform:A=LLT−R.BasedonanILUfactorizationalinearstationarymethodisdefinedbyN=(LU)−1(andM=I−NA),theILUiteration.Wethushaveanexpansionoftheoldmethodofiterativerefinement.UsingC=N−1=LUforthepreconditioning,thecomplexityoftheauxiliarysystemsdependsonthechoiceofthematrixpatternE.Ingeneral,thefollowingisrequired:E
:=(i,j)a=0,i,j=1,...,m⊂E,(i,i)i=1,...,m⊂E.ij(5.80) 2325.IterativeMethodsforSystemsofLinearEquationsTherequirementofequalityE
=Eismostoftenused.Then,andalsointhecaseoffixedexpansionsofE
,itisensuredthatforasequenceofsystemsofequationswithamatrixAthatissparseinthestrictsense,thiswillalsoholdforLandU.Allinall,onlyO(m)operationsarenecessary,includingthecalculationofLandU,asinthecaseoftheSSORpreconditioningfortheauxiliarysystemofequations.Ontheotherhand,theremainderRshouldberathersmallinordertoensureagoodconvergenceoftheILUiterationandalsotoensureasmallspectralconditionnumberκ(C−1A).PossiblematrixpatternsEareshown,forexample,in[28,pp.275ff.],whereamorespecificstructureofLandUisdiscussedifthematrixAiscreatedbyadiscretizationonastructuredgrid,forexamplebyafinitedifferencemethod.Thequestionoftheexistence(andstability)ofanILUfactorizationremainstobediscussed.Itisknownfrom(2.56)thatalsofortheexistenceofanLUfactorizationcertainconditionsarenecessary,asforexampletheM-matrixproperty.ThisisevensufficientforanILUfactorization.Theorem5.17LetA∈Rm,mbeanM-matrix.Thenforagivenpat-ternEthatsatisfies(5.80),anILUfactorizationexists.TheherebydefineddecompositionofAasA=LU−Risregularinthefollowingsense:(LU)−1≥0,(R)≥0foralli,j=1,...,m.ijijProof:See[16,p.235].
AnILU(orIC)factorizationcanbedefinedbysolvingtheequations(5.78)forlijanduijinanappropriateorder.Alternatively,theeliminationorCholeskymethodcanbeusedinitsoriginalformonthepatternE.AnimprovementoftheeigenvaluedistributionofC−1AissometimespossiblebyusinganMICfactorization(modifiedincompleteCholeskyfac-torization)insteadofanICfactorization.Incontrastto(5.79)theupdatesintheeliminationmethodforpositionsoutsidethepatternarenotignoredherebuthavetobeperformedforthecorrespondingdiagonalelement.ConcerningthereductionoftheconditionnumberbytheILUorICpreconditioningforthemodelproblem,wehavethesamesituationasfortheSSORpreconditioning.Inparticular(5.77)holds,too.TheauxiliarysystemofequationswithC=N−1,whichmeansthath(k+1)=Ng(k+1),canalsobeinterpretedasaniterationstepoftheiterationmethoddefinedbyNwithinitialvaluez(0)=0andright-handsideg(k+1).Anexpansionofthediscussedpossibilitiesforpreconditioningisthereforeobtainedbyusingafixednumberofiterationstepsinsteadofonlyone. 5.4.KrylovSubspaceMethodsforNonsymmetricSystemsofEquations233Exercises5.7LetA,A,...,A,C,C,...,C∈Rm,mbesymmetricpositive12k12ksemidefinitematriceswiththepropertyTTTmaxCix≤xAix≤bxCixforx∈R,i=1,...,kand01byµstepsofamultigriditerationonlevell−1foraand˜bandforthestartapproximation0.Setx(k+2/3)=x(k+1/3)+P−1v¯.llll−1(3)Aposteriorismoothing:Performν2smoothingsteps(k+1)ν2(k+2/3)x=Sx,lllwithν2∈{1,2,...}fixed.Table5.6.(k+1)thstepofthemultigriditerationonlevellforbilinearformaandlinearformb.andthusagaintotheGalerkindiscretizationofavariationalequationwithVl−1insteadofV,withthesamebilinearformandwithalinearformdefinedby(k+1/2)w→b(w)−aul,wforw∈Vl−1.Hencewecanignoretheassumptionofsymmetryforthebilinearformaandfindtheapproximativesolutionoftheerrorequation(5.91)ongridlevell−1bysolvingthevariationalequation(5.94).Theequivalentsystemofequationswillbederivedinthefollowing.Ontheonehand,thisprob-lemhasalowerdimensionthantheoriginalproblem,butitalsomustbesolvedforeachiteration.Thissuggeststhefollowingrecursiveprocedure:Ifwehavemorethantwogridlevels,weagainapproximatethisvariationalequationbyµmultigriditerations;inthesamewaywetreattheherebycreatedGalerkindiscretizationonlevell−2untillevel0isreached,wherewesolveexactly.Furthermore,toconcludeeachiterationstepsmoothingstepsshouldbeperformed.Thisleadstothealgorithmofthemultigriditeration.The(k+1)thstepofthemultigriditerationonlevellforthe(k)bilinearforma,linearformb,andstartingiterationxisdescribedinlTable5.6. 2445.IterativeMethodsforSystemsofLinearEquationsIngeneral,ν1=ν2isused.Inaconvergenceanalysisitturnsoutthatonlythesumofsmoothingstepsisimportant.Despitetherecursivedefi-nitionofamultigriditerationwehavehereafinitemethod,becausethelevel0isreachedafteratmostlrecursions,wheretheauxiliaryproblemwillbesolvedexactly.Forµusuallyonlythevaluesµ=1orµ=2areused.ThetermsV-cycleforµ=1andW-cycleforµ=2arecommonlyused,becauseforaniteration,thesequenceoflevelsonwhichoperationsareexecutedhavetheshapeoftheseletters(seeFigure5.6).forl=2:Levelµ=1µ=22oooo1ooooo0oooforl=3:Level3oooo2ooooo1oooooooo0oooooFigure5.6.GridlevelsfortheV-cycle(µ=1)andtheW-cycle(µ=2).Theproblemsin(5.94)and(5.95)(seeTable5.6)havetheforma(u+v,w)=b(w)forallw∈Vl−1,(5.96)wherev∈Vl−1isunknownandu∈Vlisknown.Anequivalentsystemofl−1equationsarisesbyinsertingthebasisfunctionsϕj,j=1,...,Ml−1,forwandanappropriaterepresentationforv.Ifweagaintakethel−1representationwithrespecttoϕj,wegetasin(2.34)−1Al−1Pl−1v=dl−1.(5.97)Heretheresiduald∈RMkofuonthedifferentlevelsk=0,...,liskdefinedbykkdk,i:=bϕi−au,ϕi,i=1,...,Mk. 5.5.MultigridMethod245Wenowdevelopanalternativerepresentationfor(5.97)andthecoarsegridcorrectionforpossiblegeneralizationsbeyondtheGalerkinapproxima-tions.Therefore,letR∈RMl−1,Mlbethematrixthatarisesthroughthel−1uniquerepresentationofthebasisfunctionsϕjwithrespecttothebasisϕl,whichmeanstheelementsrofRaredeterminedbytheequationsijiMlϕl−1=rϕl,j=1,...,M.jjiil−1i=1Then(5.96)isequivalenttoa(v,w)=b(w)−a(u,w)forallw∈Vl−1Ml−1⇔aP−1vϕl−1,ϕl−1=bϕl−1−au,ϕl−1,j=1,...,Ml−1ssjjjl−1s=1Ml−1MlMlMl⇔P−1varϕl,rϕl=rbϕl−au,ϕll−1ssttjiijiiis=1t=1i=1i=1Ml−1Ml⇔raϕl,ϕlrP−1v=(Rd),j=1,...,M.jitistl−1sljl−1s=1i,t=1HencethesystemofequationshastheformRARTP−1v=Rd.(5.98)ll−1lThematrixRiseasytocalculateforanode-basedbasisϕlsatisfyingiϕlal=δ,sinceinthiscasewehaveforv∈V,ijijlMlllv=vaiϕi,i=1andthereforeinparticular,Mlϕl−1=ϕl−1alϕljjiii=1andthusr=ϕl−1al.jijiForthelinearansatzinonespacedimensionwithDirichletboundaryconditions(i.e.,withV=H1(a,b)asbasicspace)thismeansthat01112211212R=...(5.99).11122 2465.IterativeMethodsforSystemsofLinearEquationsTherepresentation(5.98)canalsobeinterpretedinthisway:DuetoVl−1⊂VltheidentifydefinesanaturalprolongationfromVl−1toVl,whichmeansthatp˜:Vl−1→Vl,v→v,asillustratedbyFigure5.7.111122Figure5.7.Prolongation.ThisprolongationcorrespondstoaprolongationpfromRMl−1toRMl,thecanonicalprolongation,throughthetransitiontotherepresentationvectors(5.90).Itisgivenby−1p:=PlPl−1,(5.100)sinceforx∈RMl−1,pcanbecomposedasfollows:l−1p˜−1xl−1→Pl−1xl−1→Pl−1xl−1→PlPl−1xl−1.Obviously,pislinearandcanbeidentifiedwithitsmatrixrepresentationinRMl,Ml−1.Thenp=RT(5.101)holds,becauseMl−1MlMl−1Py=yϕl−1=yrϕl,l−1jjjjiij=1i=1j=1i.e.,RTy=P−1(Py)foranyy∈RMl−1.ll−1InthefollowingRMlwillbeendowedwithascalarproduct·,·(l),whichisanEuclideanscalarproductscaledbyafactorSl,Ml(l)xl,yl:=Slxl,iyl,i.(5.102)i=1Thescalingfactoristobechosensuchthatfortheinducednorm·landtheL2(Ω)-normonV,lC1Plxl0≤xll≤C2Plxl0(5.103)forx∈RMl,l=0,1,...,withconstantsC,Cindependentofl:Ifthe12triangulationsaremembersofaregularandquasi-uniformfamilyTh(see 5.5.MultigridMethod247Definition3.28),thenindspacedimensionsonecanchooseS=hd,withllhlbeingthemaximaldiameterofK∈Tl(seeTheorem3.43).Letr:RMl→RMl−1bedefinedbyr=p∗,(5.104)withtheadjointp∗definedwithrespecttothescalarproducts·,·(l−1)and·,·(l);thatis,34(l−1)3∗4(l−1)34(l)rxl,yl−1=pxl,yl−1=xl,pyl−1.Ifpisthecanonicalprolongation,thenriscalledthecanonicalrestriction.Fortherepresentationmatrices,Sl−1Tr=p=R.(5.105)SlInexample(5.102)ford=2withhl=hl−1/2wehaveSl−1/Sl=1/4.DuetoPp=P,thecanonicalrestrictionofRMlonRMl−1satisfiesll−1rRl=Rl−1,whereR:V→RMlisdefinedastheadjointofP,lllPx,v=x,Rv(l)forallx∈RMl,v∈V,lll0llllllbecauseforanyy∈RMl−1andforv∈V⊂V,l−1l−1l−1l34(l−1)34(l)34rRlvl−1,yl−1=Rlvl−1,pyl−1=vl−1,Plpyl−103434(l−1)=vl−1,Pl−1yl−10=Rl−1vl−1,yl−1.Using(5.105)weseetheequivalenceofequation(5.98)to(rAlp)yl−1=rdl.(5.106)Settingv:=Pl−1y˜l−1foraperhapsonlyapproximativesolutiony˜l−1of−1(5.106),thecoarsegridcorrectionwillbefinishedbyadditionofPv.Duelto−1−1−1−1Plv=PlPl−1Pl−1v=pPl−1v,thecoarsegridcorrectionis(k+2/3)(k+1/3)xl=xl+p(y˜l−1).Theabove-mentionedfactssuggestthefollowingstructureofageneralmultigridmethod:Fordiscretizationsdefiningahierarchyofdiscreteproblems,Alxl=bl,oneneedsprolongationsp:RMk−1→RMk 2485.IterativeMethodsforSystemsofLinearEquationsandrestrictionsr:RMk→RMk−1fork=1,...,landthematricesA˜k−1fortheerrorequations.Thecoarsegridcorrectionsteps(5.93)and(5.95)hencetakethefollowingform:Solve(withµstepsofthemultigridmethod)A˜y=rb−Ax(k+1/3)l−1l−1lllandset(k+2/3)(k+1/3)xl=xl+pyl−1.TheabovechoiceA˜l−1=rAlpiscalledtheGalerkinproduct.ForGalerkinapproximationsthiscoincideswiththediscretizationmatrixofthesametypeonthegridoflevell−1dueto(5.97).ThisisalsoacommonchoiceforotherdiscretizationsandthenanalternativetotheGalerkinproduct.Inviewofthechoiceofpandrweshouldobservethevalidityof(5.104).Aninterpolationaldefinitionoftheprolongationonthebasisof(finiteelement)basisfunctionsasforexample(5.101)(seealsoexample(5.99))isalsocommoninotherdiscretizations.Inmoredifficultproblems,asforexamplethosewith(dominant)convectioninadditiontodiffusivetransportprocesses,nonsymmetricproblemsarisewithasmallconstantofV-ellipticity.Heretheuseofmatrix-dependent,thatmeansAl-dependent,prolongationsandrestrictionsisrecommended.5.5.3EffortandConvergenceBehaviourInordertojudgetheefficiencyofamultigridmethodthenumberofopera-tionsperiterationandthenumberofiterations(requiredtoreachanerrorlevelε,see(5.4))hastobeestimated.Duetotherecursivestructure,thefirstnumberisnotimmediatelyclear.TheaimistohaveonlytheoptimalamountofO(Ml)operationsforsparsematrices.Forthisthedimensionsoftheauxiliaryproblemshavetodecreasesufficiently.Thisisexpressedbythefollowing:ThereexistsaconstantC>1suchthatMl−1≤Ml/Cforl∈N.(5.107)Henceweassumeaninfinitehierarchyofproblemsand/orgrids,whichalsocorrespondstotheasymptoticpointofviewofadiscretizationfromSection3.4.Relation(5.107)isthusaconditionforarefinementstrategy.ForthemodelproblemoftheFriedrichs–Kellertriangulationofarectangle(seeFigure2.9)inthecaseofaregular“red”refinementwehavehl=hl−1/2.ThusC=4,andforanalogousconstructionsindspacedimensions 5.5.MultigridMethod249C=2d.Thematricesthatappearshouldbesparse,sothatforlevellthefollowingholds:smoothingstep=CSMloperations,errorcalculationandrestrictions=CDMloperations,prolongationandcorrection=CCMloperations.Thenwecanprovethefollowing(see[16,p.326]):Ifthenumberµofmultigridstepsintherecursionsatisfiesµ0:lSlAl.ν(2)Approximationproperty:ThereexistsC>0:A−1−pA−1r≤CA−1.(5.111)All−1AlDuetoM=A−1−pA−1rASν,TGMll−1llwecanconcludethatM≤A−1−1Aν≤CSCATGMl−pAl−1rlSl,νwhichmeansthatforsufficientlylargeν,MTGM≤<1withindependentofl.Thesmoothingpropertyisofanalgebraicnature,butfortheproofoftheapproximationpropertywewilluse—atleastindirectly—theoriginalvariationalformulationoftheboundaryvalueproblemandthecorrespond-ingerrorestimate.Therefore,wediscussonlythesmoothingpropertyfor,asanexample,therelaxedRichardsonmethodforasymmetricpositivedefinitematrixAl,i.e.,11Sl=Il−ωAlwithω∈0,.λmax(Al)Let{z}MlbeanorthonormalbasisofeigenvectorsofA.Foranyinitialii=1lvectorx(0)representedinthisbasisasx(0)=Mlczitfollowsthati=1ii(compare(5.68))MlMlASνx(0)2=λ2(1−λω)2νc2=ω−2(λω)2(1−λω)2νc2lliiiiiii=1i=1562Ml≤ω−2maxξ(1−ξ)νc2.iξ∈[0,1]i=1Thefunctionξ→ξ(1−ξ)νhasitsmaximumatξ=(ν+1)−1;thusmaxνν+1ν111ν1ξmax(1−ξmax)=1−=≤.ν+1ν+1νν+1eν 5.6.NestedIterations251Hence1ASνx(0)≤x(0),llωeνwhichimplies1ASν≤.llωeνSincetheinclusionω∈(0,1/λmax(Al)]canbewrittenintheformω=σ/Alwithσ∈(0,1],wehaveCS=1/(σe).Theapproximationpropertycanbemotivatedinthefollowingway.ThefinegridsolutionxlofAlxl=dlisreplacedinthecoarsegridcorrection−1bypxl−1fromAl−1xl−1=dl−1:=rdl.Therefore,pxl−1≈Aldlshouldhold.Theformulation(5.111)thusisjustaquantitativeversionofthisrequirement.SinceinthesymmetriccaseA−1issimplythereciprocallvalueofthelargesteigenvalue,(3.140)inTheorem3.45establishestherelationtothestatementsofconvergenceinSection3.4.Foramoreexactanalysisofconvergenceandamoreextensivedescriptionofthistopicwerefertothecitedliterature(seealso[17]).Exercises5.12Determinetheprolongationandrestrictionaccordingto(5.101)and(5.104)forthecaseofalinearansatzonaFriedrichs–Kellertriangulation.5.13Provetheconsistencyofthetwo-gridmethod(5.110)inthecaseoftheconsistentsmoothingproperty.5.6NestedIterationsAsinSection5.5weassumethatbesidesthesystemofequationsAlxl=blwithMlunknowns,therearegivenanalogouslow-dimensionalsystemsofequationsAkxk=bk,k=0,...,l−1,(5.112)withMkunknowns,whereM00.Here·isanormonthebasicspaceV,andtheconstantCAgenerallydependsonthesolutionu 2525.IterativeMethodsforSystemsofLinearEquationsofthecontinuousproblem.ThediscretizationparameterhldeterminesthedimensionM:Inthesimplestcaseofauniformrefinement,hd∼1/Mlllholdsindspacedimensions.Onemayalsoexpectthatforthediscretesolution,px−x≤CChα,k=1,...,l,k−1kk1AkholdswithaconstantC>0.Here·isanormonRMk,andthe1kmappingp=p:RMk−1→RMkisaprolongation,forexamplethek−1,kcanonicalprolongationintroducedinSection5.5.Inthiscasetheestimatecanberigorouslyprovenwiththedefinitionofthecanonicalprolongation−1p=PkPk−1:px−x=P−1Px−Pxk−1kkkk−1k−1kkk≤P−1Px−PxkL[V,RMk]k−1k−1kkk≤P−1Chα+Chα≤CChαkL[Vk,RMk]AkAk−11AkwithαP−1hj−1C1=maxj=1,...,ljL[Vj,RMj]1+h.jLetthesystemofequationsbesolvedwithaniterativemethodgivenbythefixed-pointmappingΦk,k=0,...,l,whichmeansthatxkaccordingto(5.112)satisfiesxk=Φk(xk,bk).Thenitissufficienttodetermineaniteratex˜lwithanaccuracyx˜−x≤C˜hα(5.113)lllAlwithC˜A:=CA/PlL[RMl,V],becausethenwealsohavex˜αPll−Plxl≤CAhl.Ifonedoesnothaveagoodinitialiteratefromtheconcretecontext,thealgorithmofnestediterationsexplainedinTable5.7canbeused.Itisindeedafiniteprocess.Thequestionishowtochoosetheiterationnumbersmksuchthat(5.113)finallyholds,andwhetherthearisingoveralleffortisacceptable.Ananswertothisquestionisprovidedbythefollowingtheorem:Theorem5.19LettheiterativemethodΦkhavethecontractionnumberkwithrespectto·k.AssumethatthereexistconstantsC2,C3>0suchthatpL[RMk−1,RMk]≤C2,hk−1≤C3hk,forallk=1,...,l.Iftheiterationnumbersmkforthenestediterationsarechoseninsuchawaythatmk≤1/(CCα+CP),(5.114)k231l 5.6.NestedIterations253Choosemk,k=1,...,l.Letx˜0beanapproximationofx0,−1forexamplex˜0=x0=A0b0.Fork=1,...,l:(0)x˜k:=px˜k−1.Performmkiterations:(i)(i−1)x˜k:=Φkx˜k,bk,i=1,...,mk.(mk)Setx˜k:=x˜k.Table5.7.NestedIteration.thenαx˜k−xkk≤C˜Ahk,forallk=1,...,l,providedthatthisestimateholdsfork=0.Proof:Theproofisgivenbyinductiononk.Assumethattheassertionistruefork−1.Thisinducesx˜−x≤mkpx˜−xkkkkk−1kk≤mk(p(x˜−x)+px−x)kk−1k−1kk−1kk≤mkCC˜hα+CChαk2Ak−11Ak≤mk(CCα+CP)C˜hα.k231lAk
Theorem5.19allowsthecalculationofthenecessarynumberofiterations−1fortheinneriterationfromthenormspL[RMk−1,RMk],PkL[Vk,RMk]andtheconstantshk−1fork=1,...,l,aswellastheorderofconvergenceαhkofthediscretization.Inordertoestimatethenecessaryeffortaccordingto(5.114)moreex-actly,thedependenceofkofkmustbeknown.Inthefollowingweconsideronlythesituation,knownasthemultigridmethod,ofamethodofoptimalcomplexityk≤<1.Here,incontrasttoothermethods,thenumberofiterationscanbecho-senconstant(mk=mforallk=1,...,l).If,furthermore,theestimate(5.107)holdswiththeconstantC,thenanalogouslytotheconsiderationinSection5.5thetotalnumberofoperationsforthenestediterationcan 2545.IterativeMethodsforSystemsofLinearEquationsbeestimatedbyCmCMl.C−1HereCMkisthenumberofoperationsforaniterationwiththeiterationmethodΦk.InthemodelproblemoftheFriedrichs–KellertriangulationwithuniformrefinementwehaveC/(C−1)=4/3andC3=2.For·=·0asbasicnorm,α=2isatypicalcaseaccordingtoTheorem3.37.TheexistenceoftheconstantC2willherebyfinallybeensuredconsistentlybythecondition(5.103),observing(5.100).AssumingalsothattheconstantsC1,C2,Plare“small”andtheiterationmethodhasa“small”contractionnumber,onlyasmallnumberofiterationsmisnecessary,intheidealcasem=1.Atleastinthissituationwecancountononlyasmallincreaseofthenecessaryeffortthroughtheprocessofnestediterations,whichprovidesan“appropriate”approximationx˜konalllevelskofdiscretization.Finally,itistobeobservedthatthesequenceofthediscreteproblemshastobedefinedonlyduringtheprocessofthenestediteration.ThisoffersthepossibilitytocombineitwithaposteriorierrorestimatorsasdiscussedinSection4.2,inordertodevelopagridTk+1onwhichthediscreteproblemoflevelk+1isdetermined,onthebasisofx˜kasarefinementofTk. 6TheFiniteVolumeMethodFinitevolumemethodsarewidelyappliedwhendifferentialequationsindivergenceform(cf.Section0.5)ordifferentialequationsinvolvingsuchdifferentialexpressions(forexample,parabolicdifferentialequations)aretobesolvednumerically.Intheclassofsecond-orderlinearellipticdifferentialequations,expressionsoftheformLu:=−∇·(K∇u−cu)+ru=f(6.1)aretypical(cf.(0.33)),whered,ddK:Ω→R,c:Ω→R,r,f:Ω→R.Thecorresponding“parabolicversion”is∂u+Lu=f∂tandwillbetreatedinChapter7.First-orderpartialdifferentialequationssuchastheclassicalconservationlaws∇·q(u)=0,whereq:R→Rdisanonlinearvectorfielddependingonu,orhigher-orderpartialdifferentialequations(suchasthebiharmonicequation(3.36)),orevensystemsofpartialdifferentialequationscanbesuccessfullydiscretizedbythefinitevolumemethod.Incorrespondencetothecomparativelylargeclassofproblemsthatcanbetreatedbythefinitevolumemethod,thereareratherdifferentsources 2566.FiniteVolumeMethod1960ForsytheandWasowcomputationofneutrondiffusion1961Marˇcukcomputationofnuclearreactors1971McDonaldfluidmechanics1972MacCormackandPaullayfluidmechanics1973RizziandInouyefluidmechanicsin3D1977Samarskiintegro-interpolationmethod,balancemethod...1979Jamesonfinitevolumemethod1984Heinrichintegro-balancemethod,generalizedfinitedifferencemethod...1987BankandRoseboxmethod...Table6.1.Somesourcesofthefinitevolumemethod.originatingmainlyfrompracticalapplications.SomeofthesesourcesarelistedinTable6.1.Incontrasttofinitedifferenceorfiniteelementmethods,thetheoreticalunderstandingofthefinitevolumemethodremainedatanearlystageforalongtime;onlyinrecentyearshasessentialprogressbeennoted.Thefinitevolumemethodcanbeviewedasadiscretizationmethodofitsownright.Itincludesideasfrombothfinitedifferenceandfiniteelementmethods.Sointheliteratureapproachescanbefoundthatinterpretitasa“generalizedfinitedifferencemethod”orratherasavariantofthefiniteelementmethod.Inthischapter,wewillconsideronlyequationsofthetype(6.1).6.1TheBasicIdeaoftheFiniteVolumeMethodNowwewilldescribethefundamentalstepsinthederivationofthefinitevolumemethod.Forsimplicity,werestrictourselvestothecased=2andr=0.Furthermore,wesetq(u):=−K∇u+cu.Thenequation(6.1)becomes∇·q(u)=f.(6.2)Inordertoobtainafinitevolumediscretization,thedomainΩwillbesubdividedintoMsubdomainsΩisuchthatthecollectionofallthosesubdomainsformsapartitionofΩ,thatis:(1)eachΩiisanopen,simplyconnected,andpolygonallyboundedsetwithoutslits, 6.1.Basics257(2)Ωi∩Ωj=∅(i=j),(3)∪MΩ=Ω.i=1iThesesubdomainsΩiarecalledcontrolvolumesorcontroldomains.Withoutgoingintomoredetailwementionthattherealsoexistfinitevolumemethodswithawell-definedoverlappingofthecontrolvolumes(thatis,condition2isviolated).Thenextstep,whichisincommonwithallfinitevolumemethods,con-sistsinintegratingequation(6.2)overeachcontrolvolumeΩi.Afterthat,Gauss’sdivergencetheoremisapplied:

ν·q(u)dσ=fdx,i∈{1,...,M},∂ΩiΩiwhereνdenotestheouterunitnormalto∂Ωi.Bythefirstconditionofthepartition,theboundary∂Ωiisformedbystraight-linesegmentsΓij(j=1,...,ni),alongwhichthenormalν|Γij=:νijisconstant(seeFigure6.1).Sothelineintegralcanbedecomposedintoasumoflineintegralsfromwhichthefollowingequationresults:ni

νij·q(u)dσ=fdx.(6.3)j=1ΓijΩiννi2i1Ωiνi3νi5νi4Figure6.1.Acontrolvolume.Nowtheintegralsoccurringin(6.3)havetobeapproximated.Thiscanbedoneinverydifferentways,andsodifferentfinaldiscretizationsareobtained.Ingeneral,finitevolumemethodscanbedistinguishedbythefollowingcriteria:(1)thegeometricshapeofthecontrolvolumesΩi,(2)thepositionoftheunknowns(“problemvariables”)withrespecttothecontrolvolumes, 2586.FiniteVolumeMethod(3)theapproximationoftheboundary(line(d=2)orsurface(d=3))integrals.Especiallythesecondcriteriondividesthefinitevolumemethodsintotwolargeclasses:thecell-centredandthecell-vertexfinitevolumemethods.Inthecell-centredmethods,theunknownsareassociatedwiththecontrolvolumes(forexample,anycontrolvolumecorrespondstoafunctionvalueatsomeinteriorpoint(e.g.,atthebarycentre)).Inthecell-vertexmethods,theunknownsarelocatedattheverticesofthecontrolvolumes.Sometimes,insteadofthefirst-mentionedclassasubdivisionintotwoclasses,theso-calledcell-centredandnode-centredmethods,isconsidered.Thedifferenceiswhethertheproblemvariablesareassignedtothecontrolvolumesor,giventheproblemvariables,associatedcontrolvolumesaredefined.Example6.1ConsiderthehomogeneousDirichletproblemforthePois-sonequationontheunitsquare:−∆u=finΩ=(0,1)2,u=0on∂Ω.Problemvariables:aj2Functionvaluesatthenodesaiofasquaregridwithmeshwidthaj3aaij1h>0aj4Controlvolumes:Ω:={x∈Ω:|x−a|0ΩiControlvolumes:SubsquaresofthegridFigure6.3.Problemvariablesandcontrolvolumesinacell-vertexfinitevolumemethod. 2606.FiniteVolumeMethodIntheinteriorofΩ,theresultingdiscretizationyieldsa12-pointstencil(intheterminologyoffinitedifferencemethods).Remark6.3Inthefinitevolumediscretizationofsystemsofpartialdif-ferentialequations(resultingfromfluidmechanics,forexample),bothmethodsareusedsimultaneouslyfordifferentvariables;seeFigure6.4.OOOOO.....:problemvariableoftype1O.O.O.O.OO:problemvariableoftype2OOOOO....O.O.O.O.OOOOOOFigure6.4.Finitevolumediscretizationofsystemsofpartialdifferentialequations.AssetsandDrawbacksoftheFiniteVolumeMethodAssets:•FlexibilitywithrespecttothegeometryofthedomainΩ(asinfiniteelementmethods).•Admissibilityofunstructuredgrids(asinfiniteelementmethods,importantforadaptivemethods).•Simpleassembling.•Conservationofcertainlawsvalidforthecontinuousproblem(forexample,conservationlawsormaximumprinciples).Thispropertyisimportantinthenumericalsolutionofdifferentialequa-tionswithdiscontinuouscoefficientsorofconvection-dominateddiffusion-convectionequations(seeSection6.2.4).•Easylinearizationofnonlinearproblems(simplerthaninfiniteelementmethods(Newton’smethod)).•Simplediscretizationofboundaryconditions(asinfiniteelementmethods,especiallya“natural”treatmentofNeumannormixedboundaryconditions).•Inprinciple,norestrictionofthespatialdimensiondofthedomainΩ. 6.1.Basics261Drawbacks:•Smallerfieldofapplicationsincomparisonwithfiniteelementorfinitedifferencemethods.•Difficultiesinthedesignofhigherordermethods(noso-calledp-versionavailableasinthefiniteelementmethod).•Inhigherspatialdimensions(d≥3),theconstructionofsomeclassesortypesofcontrolvolumesmaybeacomplextaskandthusmayleadtoatime-consumingassembling.•Difficultmathematicalanalysis(stability,convergence,...).Exercises6.1Giventheboundaryvalueproblem−(au
)
=0in(0,1),u(0)=1,u(1)=0,withpiecewiseconstantcoefficientsκα,x∈(0,ξ),a(x):=α,x∈(ξ,1),whereα,κarepositiveconstantsandξ∈(0,1)Q:(a)Whatistheweaksolutionu∈H1(0,1)ofthisproblem?(b)Forgeneral“smooth”coefficientsa,thedifferentialequationisobviouslyequivalentto−au

−a
u
=0.Therefore,thefollowingdiscretizationissuggested:ui−1−2ui+ui+1ai+1−ai−1ui+1−ui−1−ai−=0,h22h2hwhereanequidistantgridwiththenodesxi=ih(i=0,...,N+1)andai:=a(xi),ui:≈u(xi)isused.Thisdiscretizationisalsoformallycorrectinthegivensituationofdiscontinuouscoefficients.Findthediscretesolution(u)Ninthisii=1case.(c)Underwhatconditionsdothevaluesuiconvergetou(xi)forh→0? 2626.FiniteVolumeMethod6.2TheFiniteVolumeMethodforLinearEllipticDifferentialEquationsofSecondOrderonTriangularGridsInthissectionwewillexplainthedevelopmentandtheanalysisofafinitevolumemethodof“cell-centred”typeforamodelproblem.Here,Ω⊂R2isabounded,simplyconnecteddomainwithapolygonalboundary,butwithoutslits.6.2.1AdmissibleControlVolumesTheVoronoiDiagramBy{ai}i∈Λ⊂ΩwedenoteaconsecutivelynumberedpointsetthatincludesallverticesofΩ,whereΛisthecorrespondingsetofindices.Typically,thepointsaiareplacedatthosepositionswherethevaluesu(ai)oftheexactsolutionuaretobeapproximated.TheconvexsetΩ˜:=x∈R2|x−a|<|x−a|forallj=iiijiscalledtheVoronoipolygon(orDirichletdomain,Wigner–Seitzcell,Thiessenpolygon,...).ThefamilyΩ˜iiscalledtheVoronoidiagrami∈Λofthepointset.{ai}i∈Λ.....boundaryofΩ.∼..boundaryofΩiFigure6.5.Voronoidiagram.TheVoronoipolygonsareconvex,butnotnecessarilybounded,sets(con-siderthesituationneartheboundaryinFigure6.5).Theirboundariesarepolygons.TheverticesofthesepolygonsarecalledVoronoivertices.ItcanbeshownthatatanyVoronoivertexatleastthreeVoronoipoly-gonsmeet.Accordingtothisproperty,VoronoiverticesareclassifiedintoregularanddegenerateVoronoivertices:InaregularVoronoivertex,theboundariesofexactlythreeVoronoipolygonsmeet,whereasadegenerateVoronoivertexissharedbyatleastfourVoronoipolygons.Inthelattercase,allthecorrespondingnodesaiarelocatedatsomecircle(theyare“cocyclic”,cf.Figure6.6). 6.2.FiniteVolumeMethodonTriangularGrids263.a2a.a(a1-a4arecocyclic)3.5.a.a14Figure6.6.DegenerateandregularVoronoivertex.NowtheelementsΩi(controlvolumes)ofthepartitionofΩrequiredforthedefinitionofthefinitevolumemethodcanbeintroducedasfollows:Ωi:=Ω˜i∩Ω,i∈Λ.Asaconsequence,thedomainsΩineednotnecessarilybeconvexifΩisnonconvex(cf.Figure6.5).Furthermore,thefollowingnotationwillbeused:Λi:=j∈Λ{i}:∂Ωi∩∂Ωj=∅,i∈Λ,forthesetofindicesofneighbouringnodes,Γij:=∂Ωi∩∂Ωj,j∈Λi,forajointpieceoftheboundariesofneighbouringcontrolvolumes,mijforthelengthofΓij.ThedualgraphoftheVoronoidiagramisdefinedasfollows:Anypairofpointsai,ajsuchthatmij>0isconnectedbyastraight-linesegment.Inthisway,afurtherpartitionofΩwithaninterestingpropertyresults.Theorem6.4IfallVoronoiverticesareregular,thenthedualgraphcoin-cideswiththesetofedgesofatriangulationoftheconvexhullofthegivenpointset.ThistriangulationiscalledaDelaunaytriangulation.IfamongtheVoronoiverticestherearedegenerateones,thenatri-angulationcanbeobtainedfromthedualgraphbyasubsequentlocaltriangulationoftheremainingm-polygons(m≥4).ADelaunaytriangula-tionhastheinterestingpropertythattwointerioranglessubtendedbyanygivenedgesumtonomorethanπ.InthisrespectDelaunaytriangulationssatisfythefirstpartoftheangleconditionformulatedinSection3.9forthemaximumprincipleinfiniteelementmethods.Therefore,ifΩisconvex,thenweautomaticallygetatriangulationto-getherwiththeVoronoidiagram.InthecaseofanonconvexdomainΩ,certainmodificationscouldberequiredtoachieveacorrecttriangulation. 2646.FiniteVolumeMethod.....Thisedgehastobe.removedfromthe..Delaunaytriangulation.Figure6.7.DelaunaytriangulationtotheVoronoidiagramfromFigure6.5.TheimplicationVoronoidiagram⇒Delaunaytriangulation,whichwehavejustdiscussed,suggeststhatweaskabouttheconversestatement.Wedonotwanttoansweritcompletelyatthispoint,butwegivethefollowingsufficientcondition.Theorem6.5IfaconformingtriangulationofΩ(inthesenseoffiniteelementmethods)consistsofnonobtusetrianglesexclusively,thenitisaDelaunaytriangulation,andthecorrespondingVoronoidiagramcanbeconstructedbymeansoftheperpendicularbisectorsofthetriangles’edges.Wementionthatthecentreofthecircumcircleofanonobtusetriangleislocatedwithintheclosureofthattriangle.Intheanalysisofthefinitevolumemethod,thefollowingrelationisimportant.Lemma6.6GivenanonobtusetriangleKwithverticesaik,k∈{1,2,3},thenforthecorrespondingpartsΩik,K:=Ωik∩KofthecontrolvolumesΩik,wehave11|K|≤|Ωik,K|≤|K|,k∈{1,2,3}.42TheDonalddiagramIncontrasttotheVoronoidiagram,wheretheconstructionstartsfromagivenpointset,thestartingpointhereisatriangulationThofΩ,whichisallowedtocontainobtusetriangles.Again,letKbeatrianglewithverticesaik,k∈{1,2,3}.WedefineΩik,K:=x∈Kλj(x)<λk(x),j=k,whereλkdenotethebarycentriccoordinateswithrespecttoaik(cf.(3.51)).Obviously,thebarycentresatisfiesa=1(a+a+a),and(see,forS3i1i2i3comparison,Lemma6.6)3|Ωik,K|=|K|,k∈{1,2,3}.(6.4)ThisrelationisasimpleconsequenceofthegeometricinterpretationofthebarycentriccoordinatesasareacoordinatesgiveninSection3.3.The 6.2.FiniteVolumeMethodonTriangularGrids265ai2.Ωi2,K.aSΩ.aii1,K1Ωi3,K.ai3Figure6.8.ThesubdomainsΩi.k,Krequiredcontrolvolumesaredefinedasfollows(seeFigure6.8):#Ωi:=intΩi,K,i∈Λ.K:∂KaiThefamily{Ωi}i∈ΛiscalledaDonalddiagram.ThequantitiesΓij,mij,andΛiaredefinedsimilarlyasinthecaseoftheVoronoidiagram.WementionthattheboundarypiecesΓijarenotnecessarilystraight,butpolygonalingeneral.6.2.2FiniteVolumeDiscretizationThemodelunderconsiderationisaspecialcaseofequation(6.1).Insteadofthematrix-valueddiffusioncoefficientKwewilltakeascalarcoefficientk:Ω→R,thatis,K=kI.Moreover,homogeneousDirichletboundaryconditionsaretobesatisfied.Sotheboundaryvalueproblemreadsasfollows:−∇·(k∇u−cu)+ru=finΩ,(6.5)u=0on∂Ω,withk,r,f:Ω→R,c:Ω→R2.TheCaseoftheVoronoiDiagramLetthedomainΩbepartionedbyaVoronoidiagramandthecorrespond-ingDelaunaytriangulation.DuetothehomogeneousDirichletboundaryconditions,itissufficienttoconsideronlythosecontrolvolumesΩithatareassociatedwithinnernodesai∈Ω.Therefore,wedenotethesetofindicesofallinnernodesbyΛ:=i∈Λai∈Ω.Inthefirststep,thedifferentialequation(6.5)isintegratedoverthesinglecontrolvolumesΩi:


−∇·(k∇u−cu)dx+rudx=fdx,i∈Λ.(6.6)ΩiΩiΩi 2666.FiniteVolumeMethodTheapplicationofGauss’sdivergencetheoremtothefirstintegraloftheleft-handsideof(6.6)yields

∇·(k∇u−cu)dx=ν·(k∇u−cu)dσ.Ωi∂ΩiDueto∂Ωi=∪j∈ΛiΓij(cf.Figure6.9),itfollowsthat

∇·(k∇u−cu)dx=νij·(k∇u−cu)dσ,Ωij∈ΛiΓijwhereνijisthe(constant)outerunitnormaltoΓij(withrespecttoΩi).InthenextstepweapproximatethelineintegralsoverΓ.ij..ai.Γijaj.Figure6.9.TheedgeΓij.First,thecoefficientskandνij·careapproximatedonΓijbyconstantsµij>0,respectivelyγij:k|Γij≈µij=const>0,νij·c|Γij≈γij=const.Inthesimplestcase,theapproximationcanberealizedbythecorrespond-ingvalueatthemidpointaΓijofthestraight-linesegmentΓij.Abetterchoiceis
1νij·cdσ,mij>0,γij:=mijΓij(6.7)νij·c(aΓij),mij=0.Wethusobtain

∇·(k∇u−cu)dx≈[µij(νij·∇u)−γiju]dσ.ΩiΓijj∈ΛiThenormalderivativesareapproximatedbydifferencequotients;thatis,u(aj)−u(ai)νij·∇u≈withdij:=|ai−aj|.dijThisformulaisexactforsuchfunctionsthatarelinearalongthestraight-linesegmentbetweenthepointsai,aj.SoitremainstoapproximatetheintegralofuoverΓij.Forthis,aconvexcombinationofthevaluesofuat 6.2.FiniteVolumeMethodonTriangularGrids267thenodesaiandajistaken:u|Γij≈riju(ai)+(1−rij)u(aj),whererij∈[0,1]isaparametertobedefinedsubsequently.Ingeneral,rijdependsonµij,γij,anddij.Collectingalltheaboveapproximations,wearriveatthefollowingrelation:
∇·(k∇u−cu)dxΩiu(aj)−u(ai)≈µij−γij[riju(ai)+(1−rij)u(aj)]mij.dijj∈ΛiToapproximatetheremainingintegralsfrom(6.6),thefollowingformulasareused:
rudx≈r(ai)u(ai)mi=:riu(ai)mi,withmi:=|Ωi|,Ωi
fdx≈f(ai)mi=:fimi.ΩiInsteadofri:=r(ai)orfi:=f(ai),theapproximations

11ri:=rdxrespectivelyfi:=fdx(6.8)miΩimiΩicanalsobeused.Denotingtheunknownapproximatevaluesforu(ai)byui,weobtainthefollowinglinearsystemofequations:ui−ujµij+γij[rijui+(1−rij)uj]mij+riuimidij(6.9)j∈Λi=fimi,i∈Λ.Thisrepresentationclearlyindicatestheaffinityofthefinitevolumemethodtothefinitedifferencemethod.However,forthesubsequentanalysisitismoreconvenienttorewritethissystemofequationsintermsofadiscretevariationalequality.Multiplyingtheithequationin(6.9)byarbitrarynumbersvi∈Randsummingtheresultsupoveri∈Λ,wegetu−uijviµij+γij[rijui+(1−rij)uj]mij+riuimidiji∈Λj∈Λi=fivimi.i∈ΛFurther,letVhdenotethespaceofcontinuousfunctionsthatarepiecewiselinearoverthe(Delaunay)triangulationofΩandthatvanishon∂Ω.ThenthevaluesuiandvicanbeinterpolatedinVh;thatis,thereareunique 2686.FiniteVolumeMethoduh,vh∈Vhsuchthatuh(ai)=ui,vh(ai)=viforalli∈Λ.ThefollowingdiscretebilinearformsonVh×Vhcanthenbedefined:m0ijah(uh,vh):=viµij(ui−uj),diji∈Λj∈Λibh(uh,vh):=vi[rijui+(1−rij)uj]γijmij,i∈Λj∈Λidh(uh,vh):=riuivimi,i∈Λa(u,v):=a0(u,v)+b(u,v)+d(u,v).hhhhhhhhhhhhFinally,fortwocontinuousfunctionsv,w∈C(Ω),wesetw,v0,h:=wivimi,i∈Λwherevi:=v(ai),wi:=w(ai).Remark6.7·,·0,hisascalarproductonVh.Inparticular,thefollowingnormcanbeintroduced:(vh0,h:=vh,vh0,h,vh∈Vh.(6.10)In(3.136)adiscrete(L2-)normforageneralfiniteelementspacevhashbeendefinedusingthesamenotation.Thismultipleuseseemstobeaccept-able,sinceforregulartriangulationsbothnormsareequivalentuniformlyinh(seeRemark6.16below).Nowthediscretevariationalformulationofthefinitevolumemethodisthis:Finduh∈Vhsuchthatah(uh,vh)=f,vh0,hforallvh∈Vh.(6.11)Uptonow,thechoiceoftheweightingparametersrijhasremainedopen.Forthis,twocasescanberoughlydistinguished:(1)Thereexistsapairofindices(i,j)∈Λ×Λsuchthatµij|γij|dij.(2)Thereisnosuchpair(i,j)withµij|γij|dij.Inthesecondcase,anappropriatechoiceisr≡1.Tosomeextent,thisij2canbeseenasageneralizationofthecentraldifferencemethodtononuni-formgrids.Thefirstcasecorrespondstoalocallyconvection-dominatedsituationandrequiresacarefulselectionoftheweightingparametersrij.ThiswillbeexplainedinmoredetailinSection9.3.Ingeneral,theweightingparametersareofthefollowingstructure:γijdijrij=R,(6.12)µij 6.2.FiniteVolumeMethodonTriangularGrids269γijdijwhereR:R→[0,1]issomefunctiontobespecified.TheargumentµijiscalledthelocalP´ecletnumber.TypicalexamplesforthisfunctionRare1R(z)=[sign(z)+1],fullupwinding,2(1−τ)/2,z<0,2R(z)=τ(z):=max0,1−,(1+τ)/2,z≥0,|z|1zR(z)=1−1−,exponentialupwinding.zez−1Allthesefunctionspossessmanycommonproperties.Forexample,forallz∈R,(P1)[1−R(z)−R(−z)]z=0,&'(P2)R(z)−1z≥0,(6.13)2(P3)1−[1−R(z)]z≥0.NotethattheconstantfunctionR=1satisfiestheconditions(P1)and2(P2)butnot(P3).TheCaseoftheDonaldDiagramLetthedomainΩbetriangulatedasinthefiniteelementmethod.Then,followingtheexplanationsgiveninthesecondpartofSection6.2.1,thecorrespondingDonalddiagramcanbecreated.Thediscretebilinearforminthiscaseisdefinedbyah(uh,vh):=k∇uh,∇vh0+bh(uh,vh)+dh(uh,vh);thatis,theprincipalpartofthedifferentialexpressionisdiscretizedasinthefiniteelementmethod,wherebh,dh,andVharedefinedasinthefirstpartofthissection.6.2.3ComparisonwiththeFiniteElementMethodAswehavealreadyseeninExample6.1,itmayhappenthatafinitevolumediscretizationcoincideswithafinitedifferenceorfiniteelementdiscretiza-tion.WealsomentionthatthecontrolvolumesfromthatexampleareexactlytheVoronoipolygonstothegridpoints(i.e.,tothenodesofthetriangulation).Herewewillconsiderthisobservationinmoredetail.By{ϕi}i∈Λwede-notethenodalbasisofthespaceVhofcontinuous,piecewiselinearfunctionsonaconformingtriangulationofthedomainΩ.Lemma6.8LetThbeaconformingtriangulationofΩ(inthesenseoffiniteelementmethods),alltrianglesofwhicharenonobtuse,andconsiderthecorrespondingVoronoidiagraminaccordancewithTheorem6.5.Then,foranarbitrarytriangleK∈Thwithverticesai,aj(i=j),thefollowing 2706.FiniteVolumeMethodrelationholds:
Kmij∇ϕj·∇ϕidx=−,KdijwheremKisthelengthofthesegmentofΓthatintersectsK.ijijProof:HereweusesomeofthenotationandthefactspreparedatthebeginningofSection3.9.Inparticular,αKdenotestheinteriorangleofKijthatislocatedinoppositetheedgewithverticesai,aj.Next,thefollowingequalityisanobviousfactfromelementarygeometry:2sinαKmK=ijijcosαKd.Itremainstorecalltherelationijij
1K∇ϕj·∇ϕidx=−cotαijK2fromLemma3.47,andthestatementimmediatelyfollows.
Corollary6.9UndertheassumptionsofLemma6.8,wehavefork≡1,∇u,∇v=a0(u,v)forallu,v∈V.hh0hhhhhhProof:Itissufficienttoverifytherelationforvh=ϕiandarbitraryi∈Λ.First,weseethat
∇uh,∇ϕi0=∇uh·∇ϕidx.K⊂suppϕiKFurthermore,

∇uh·∇ϕidx=uj∇ϕj·∇ϕidxKKj:∂Kaj

=ui∇ϕi·∇ϕidx+uj∇ϕj·∇ϕidx.KKj=i:∂KajSince1=ϕjj:∂KajoverK,itfollowsthat∇ϕi=−∇ϕj;(6.14)j=i:∂Kajthatis,bymeansofLemma6.8,

∇uh·∇ϕidx=(uj−ui)∇ϕj·∇ϕidxKKj=i:∂Kaj 6.2.FiniteVolumeMethodonTriangularGrids271mKij=(ui−uj).(6.15)dijj=i:∂KajSummingoverallK⊂suppϕi,wegetmij0∇uh,∇ϕi0=(ui−uj)=ah(uh,ϕi).
dijj∈ΛiRemark6.10Byamoresophisticatedargumentationitcanbeshownthattheabovecorollaryremainsvalidifthediffusioncoefficientkiscon-stantonalltrianglesK∈Thandiftheapproximationµijischosenaccordingto
KK1k|Kmij+k|Kmijkdσ=,mij>0,µij:=mijΓijmij(6.16)0,mij=0,whereK,K
arebothtrianglessharingtheverticesa,a.ijTreatmentofMatrix-valuedDiffusionCoefficientsCorollary6.9andRemark6.10arevalidonlyinthespatialdimensiond=2.However,formoregeneralcontrolvolumes,higherspatialdimensions,ornotnecessarilyscalardiffusioncoefficients,weakerstatementscanbeproven.Asanexample,wewillstatethefollowingfact.Asaby-product,wealsoobtainanideaforhowtoderivediscretizationsinthecaseofmatrix-valueddiffusioncoefficients.ForabetterdistinctionbetweentheelementsKofthetriangulationandthediffusioncoefficient,wekeepthenotationkforthediffusioncoefficient,evenifkisallowedtobeamatrix-valuedfunctiontemporarily.Lemma6.11LetThbeaconformingtriangulationofΩ,whereinthecaseoftheVoronoidiagramitisadditionallyrequiredthatalltrianglesbenonobtuse.Furthermore,assumethatthediffusionmatrixk:Ω→R2,2isconstantonthesingleelementsofTh.Thenforanyi∈ΛandK∈Thwehave

(k∇uh)·∇ϕidx=−(k∇uh)·νdσforalluh∈Vh,K∂Ωi∩Kwhere{Ωi}i∈ΛiseitheraVoronoioraDonalddiagramandνdenotestheouterunitnormalwithrespecttoΩi.Withoutdifficulties,theproofcanbecarriedoverfromtheproofofarelatedresultin[20,Lemma6.1].Nowwewillshowhowtousethisfacttoformulatediscretizationsforthecaseofmatrix-valueddiffusioncoefficients.Namely,usingrelation(6.14), 2726.FiniteVolumeMethodweeasilyseethat

(k∇uh)·νdσ=uj(k∇ϕj)·νdσ∂Ωi∩K∂Ωi∩Kj:∂Kaj
=(uj−ui)(k∇ϕj)·νdσ.j=i:∂Kaj∂Ωi∩KSummingoveralltrianglesthatlieinthesupportofϕi,weobtainbyLemma6.11therelation

(k∇uh)·∇ϕidx=(ui−uj)(k∇ϕj)·νdσ.(6.17)Ω∂Ωij∈ΛiWiththedefinition
dij(k∇ϕj)·νdσ,mij>0,µij:=mij∂Ωi(6.18)0,mij=0,itfollowsthat
mij(k∇uh)·∇ϕidx=µij(ui−uj).Ωdijj∈ΛiNotethat,inthecaseofVoronoidiagrams,(6.16)isaspecialcaseofthechoice(6.18).Consequently,inordertoobtainadiscretizationforthecaseofamatrix-valueddiffusioncoefficient,itissufficienttoreplaceinthebilinearformbhand,iftheVoronoidiagramisused,alsoina0,thetermsinvolvingµhijaccordingtoformula(6.18).ImplementationoftheFiniteVolumeMethodInprinciple,thefinitevolumemethodcanbeimplementedindifferentways.Ifthelinearsystemofequationsisimplementedinanode-orientatedmanner(asinfinitedifferencemethods),theentriesofthesystemmatrixAhandthecomponentsoftheright-handsideqhcanbetakendirectlyfrom(6.9).Ontheotherhand,anelement-orientatedassemblingispossible,too.Thisapproachispreferable,especiallyinthecasewhereanexistingfiniteelementprogramwillbeextendedbyafinitevolumemodule.Theideaofhowtodothisissuggestedbyequation(6.17).Namely,foranytriangleK∈Th,therestrictedbilinearformah,Kwiththeappropriatedefinitionofµijaccordingto(6.18)isdefinedasfollows:ah,K(uh,vh):=ui−ujKKviµij+γij[rijui+(1−rij)uj]mij+riuimi,diji∈Λj=i:∂Kaj 6.2.FiniteVolumeMethodonTriangularGrids273wheremK:=|Ω∩K|.ThenthecontributionofthetriangleKtotheiimatrixentry(Ah)ijofthematrixAhisequaltoah,K(ϕj,ϕi).Inthesameway,theright-handsideof(6.9)canbesplitelementwise.6.2.4PropertiesoftheDiscretizationHerewewillgiveashortoverviewofbasicpropertiesoffinitevolumemethods.Forthesakeofsimplicity,werestrictourselvestothecaseofaconstantscalardiffusioncoefficientk>0.Then,inparticular,itisusefultosetµij:=kforalli∈Λ,j∈Λi.Lemma6.12Supposetheapproximationsγijofνij·c|Γijsatisfyγji=−γijandtherijaredefinedby(6.12)withafunctionRsatisfying(P1).Thenwegetforalluh,vh∈Vh,1bh(uh,vh)=uiviγijmij2i∈Λj∈Λi56111+rij−(ui−uj)(vi−vj)+(ujvi−uivj)γijmij.222i∈Λj∈ΛiProof:First,weobservethatbhcanberewrittenasfollows:561bh(uh,vh)=vi(1−rij)uj−−rijuiγijmij2i∈Λj∈Λi1(6.19)+uiviγijmij.2i∈Λj∈ΛiInthefirstterm,wechangetheorderofsummationandrenametheindices:561bh(uh,vh)=vj(1−rji)ui−−rjiujγjimji2i∈Λj∈Λi1+uiviγijmij.2i∈Λj∈ΛiNextwemakeuseofthefollowingrelations,whicheasilyresultfromdji=dijandtheassumptionsonγijandrij:11(1−rji)γji=−rijγij,−rjiγji=−rijγij.22Soweget,duetomji=mij,561bh(uh,vh)=vj−rijui−−rijujγijmij2i∈Λj∈Λi1+uiviγijmij.2i∈Λj∈Λi 2746.FiniteVolumeMethodTakingthearithmeticmeanofbothrepresentationsofbh,wearriveat1bh(uh,vh)=uiviγijmij25i∈Λj∈Λi611+(1−rij)ujvi−rijuivj−−rji(uivi+ujvj)γijmij22i∈Λj∈Λi511=−rij(ujvi+uivj−uivi−ujvj)22i∈Λj∈Λi611+(ujvi−uivj)γijmij+uiviγijmij.22i∈Λj∈Λi
Corollary6.13Letc1,c2,∇·c∈C(Ω).UndertheassumptionsofLemma6.12andalsoassumingproperty(P2)forR,thebilinearformbhsatisfiesforallvh∈Vhtheestimate

1b(v,v)≥v2∇·cdx+(γ−ν·c)dσ.(6.20)hhhiijij2ΩiΓiji∈Λj∈ΛiProof:Duetor−1γ≥0,becauseofproperty(P2)in(6.13),itij2ijimmediatelyfollowsthat11b(v,v)≥v2γm=v2γm.hhhiijijiijij22i∈Λj∈Λii∈Λj∈ΛiFortheinnersum,wecanwrite
γijmij=γijdσj∈Λij∈ΛiΓij

=νij·cdσ+(γij−νij·c)dσ.j∈ΛiΓijj∈ΛiΓijThefirsttermcanberewrittenasanintegralovertheboundaryofΩi,i.e.,

νij·cdσ=ν·cdσ.Γij∂Ωij∈ΛiByGauss’sdivergencetheorem,itfollowsthat

ν·cdσ=∇·cdx.∂ΩiΩi
6.2.FiniteVolumeMethodonTriangularGrids275Remark6.14Iftheapproximationsγijarechosenaccordingto(6.7),thenγji=−γij,and(6.20)simplifiesto
1b(v,v)≥v2∇·cdx.hhhi2Ωii∈ΛUsingasimilarargumentasinthetreatmentofthetermj∈ΛiγijmijintheproofofCorollary6.13,thevaluedh(vh,vh)canberepresentedasfollows:
22dh(vh,vh)=rivimi=viridxΩii∈Λi∈Λ

22=virdx+vi(ri−r)dx.(6.21)ΩiΩii∈Λi∈ΛThesecondtermvanishesiftheapproximationsriaredefinedasin(6.8).Theorem6.15Lettherijbedefinedby(6.12)withRsatisfying(P1)and(P2).Supposek>0,c,c,∇·c,r∈C(Ω),r+1∇·c≥r=const≥0on1220Ωandthattheapproximationsγij,respectivelyri,arechosenaccordingto(6.7),respectively(6.8).UndertheassumptionsofLemma6.8,wehaveforallvh∈Vh,a(v,v)≥k∇v,∇v+rv2m=k|v|2+rv2;hhhhh00iih10h0,hi∈Λthatis,thebilinearformahisVh-ellipticuniformlywithrespecttoh.Proof:Westartwiththeconsiderationofa0(v,v).DuetoCorollary6.9,hhhtherelation02ah(vh,vh)=k∇vh,∇vh0=k|vh|1holds.Furthermore,byRemark6.14andequation(6.21),wehave
1b(v,v)+d(v,v)≥v2∇·c+rdx≥rv2m.hhhhhhi0iiΩi2i∈Λi∈ΛSincebydefinition,a(v,v)=a0(v,v)+b(v,v)+d(v,v),hhhhhhhhhhhhbothrelationsyieldtheassertion.
Remark6.16Letthefamilyoftriangulations(Th)hberegular.Thenthenormsdefinedin(3.136)andin(6.10)andalsothenorms·0,hand·0areequivalentonVhuniformlywithrespecttoh;i.e.,thereexisttwoconstantsC1,C2>0independentofhsuchthatC1v0≤v0,h≤C2v0forallv∈Vh. 2766.FiniteVolumeMethodProof:DuetoTheorem3.43(i)onlytheuniformequivalenceofthedis-creteL2-normshastobeshown.Denotingsuchanequivalenceby∼=,wehaveforv∈Vhwithvi:=v(ai)fori∈Λ,1/21/2|v|2m=|v|2|Ω|iiii,Ki∈Λi∈ΛK∈Th:K∩Ωi=∅1/2∼=|v|2|K|1/2ii∈ΛK∈Th:K∩Ωi=∅duetoLemma6.6or(6.4)1/2∼=|K||v|2iK∈Thi:ai∈K1/2∼=h2|v|2,KiK∈Thi:ai∈Ksinceduetotheregularityof(Th)hthereisauniformlowerboundfortheanglesofK∈Th(see(3.93))andthusauniformupperboundonthenumberofK∈ThsuchthatK∩Ωi=∅.
Corollary6.17UndertheassumptionsofTheorem6.15andforaregularfamilyoftriangulations(Th)hthereexistsaconstantα>0independentofhsuchthat2ah(vh,vh)≥αvh1forallvh∈Vh.Proof:ByRemark6.16andTheorem6.15,a(v,v)≥k|v|2+rC2v2,hhhh101h0i.e.,wecantakeα:=min{k;rC2}.
01Theorem6.15(orCorollary6.17)assertsthestabilityofthemethod.Itisthefundamentalresultfortheproofofanerrorestimate.Theorem6.18Let{Th}h∈(0,h¯]bearegularfamilyofconformingtriangu-lations,whereinthecaseoftheVoronoidiagramitisadditionallyrequiredthatalltrianglesbenonobtuse.Furthermore,supposein(6.5)thatk>0,c,c,∇·c,r∈C(Ω),r+1∇·c≥r=const>0onΩ,f∈C1(Ω),and1220thattheapproximationsγij,respectivelyri,arechosenaccordingto(6.7), 6.2.FiniteVolumeMethodonTriangularGrids277respectively(6.8).Lettherijbedefinedby(6.12)withRsatisfying(P1)and(P2).Iftheexactsolutionuof(6.5)belongstoH2(Ω)andu∈Vhhdenotesthesolutionof(6.11),thenu−uh1≤Ch[u2+|f|1,∞],wheretheconstantC>0isindependentofh.Proof:TheproofrestsonasimilarideatothoseintheproofandtheapplicationofStrang’sfirstlemma(Theorem3.38)inSection3.6.DenotingbyIh:CΩ¯→Vhtheinterpolationoperatordefinedin(3.71)andsettingvh:=uh−Ih(u),wehaveah(vh,vh)=ah(uh,vh)−ah(Ih(u),vh)=f,vh0,h−ah(Ih(u),vh)

=f,vh0,h−vifdx+vifdx−ah(Ih(u),vh).i∈ΛΩii∈ΛΩiBythedefinitionofthediscreteformf,vh0,handbythedifferentialequation(6.5),consideredasanequationinL2(Ω),weget

ah(vh,vh)=vi(fi−f)dx+viLudx−ah(Ih(u),vh),ΩiΩii∈Λi∈ΛwhereLu=−∇·(k∇u−cu)+ru.Forf∈C1(Ω)andthechoicef:=f(a),itiseasytoseethatii|fi−f(x)|≤|f|1,∞maxhK≤Ch|f|1,∞forallx∈Ωi.K:ai∈KSoitfollowsthat
vi(fi−f)dx≤Ch|f|1,∞|vi|miΩii∈Λi∈Λ$%1/2$%1/2≤Ch|f|v2mm1,∞iiii∈Λi∈Λ!√≤|Ω|≤Ch|f|1,∞vh0,h.Fortheotherchoiceoffi(see(6.8)),thesameestimateistriviallysatisfied.Thedifficultpartoftheproofistogetanestimateoftheconsistencyerror
viLudx−ah(Ih(u),vh).Ωii∈Λ 2786.FiniteVolumeMethodThisisveryextensive,andsowewillomitthedetails.Acompleteproofofthefollowingresultisgiveninthepaper[40]:
221/2viLudx−ah(Ih(u),vh)≤Chu2|vh|1+vh0,h.(6.22)Ωii∈ΛPuttingbothestimatestogetherandtakingintoconsiderationRemark6.16,wearriveat221/2ah(vh,vh)≤Ch[u2+|f|1,∞]|vh|1+vh0,h≤Ch[u2+|f|1,∞]vh1.ByCorollary6.17,weconcludefromthisthatvh1≤Ch[u2+|f|1,∞].Itremainstoapplythetriangleinequalityandthestandardinterpolationerrorestimate(cf.Theorem3.29withk=1orTheorem3.35)u−uh1≤u−Ih(u)1+vh1≤Ch[u2+|f|1,∞].
WepointoutthattheerrormeasuredintheH1-seminormisofthesameorderasforthefiniteelementmethodwithlinearfiniteelements.Nowwewillturntotheinvestigationofsomeinterestingpropertiesofthemethod.GlobalConservativityHereweconsidertheboundaryvalueproblem−∇·(k∇u−cu)=finΩ,ν·(k∇u−cu)=gon∂Ω.IntegratingthedifferentialequationoverΩ,weconcludefromGauss’sdivergencetheoremthat


−∇·(k∇u−cu)dx=−ν·(k∇u−cu)dσ=−gdσ,Ω∂Ω∂Ωandhence

gdσ+fdx=0.∂ΩΩThisisanecessarycompatibilityconditionforthedatadescribingthebal-ancebetweenthetotalflowovertheboundaryandthedistributedsources.Wewilldemonstratethatthediscretizationrequiresadiscretizedversionofthiscompatibilitycondition,whichiscalleddiscreteglobalconservativity.Therefore,wefirsthavetodefinethediscretizationfortheabovetypeofboundaryconditions.Obviously,forinnercontrolvolumesΩi(i∈Λ),there 6.2.FiniteVolumeMethodonTriangularGrids279isnoneedforanymodifications.SowehavetoconsideronlytheboundarycontrolvolumesΩi(i∈∂Λ:=ΛΛ).InthecaseoftheVoronoidiagramwehave

−∇·(k∇u−cu)dx=−ν·(k∇u−cu)dσΩi∂Ωi

=−ν·(k∇u−cu)dσ−ν·(k∇u−cu)dσ(6.23)Γij∂Ωi∩∂Ωj∈Λi

=−ν·(k∇u−cu)dσ−gdσ.Γij∂Ωi∩∂Ωj∈ΛiSincethelineintegralsoverΓijcanbeapproximatedinthestandardway,wegetthefollowingequation:ui−ujviµij+γij[rijui+(1−rij)uj]mij(6.24)diji∈Λj∈Λi
−vigdσ=fivimi,∂Ωi∩∂Ωi∈Λi∈ΛwheretheansatzandtestspaceVhconsistsofallcontinuousfunctionsoverΩthatarepiecewiselinearwithrespecttotheunderlyingtriangulation¯(thatis,intheboundarynodesnofunctionvaluesareprescribed).Itisagainassumedthattherijaredefinedby(6.12)withafunctionRsatisfying(P1)andγji=−γij.Obviously,theparticularfunctionih:≡1belongstoVh.Soweareallowedtosetvh=ihinthediscretization.Then,repeatingtheabovesymmetryargument(cf.theproofofLemma6.12),wegetmijmijµij(ui−uj)=−µij(ui−uj),dijdiji∈Λj∈Λii∈Λj∈Λithatis,mijµij(ui−uj)=0.diji∈Λj∈ΛiOntheotherhand,usingthesameargument,wehave[rijui+(1−rij)uj]γijmiji∈Λj∈Λi=[rjiuj+(1−rji)ui]γjimjii∈Λj∈Λi=−[(1−rij)uj+rijui]γijmij.(6.25)i∈Λj∈Λi 2806.FiniteVolumeMethodConsequently,thistermvanishes,too.Becauseof

vigdσ=gdσ,∂Ωi∩∂Ω∂Ωi∈Λitfollowsthat

−gdσ=fivimi=fimi≈fdx.(6.26)∂ΩΩi∈Λi∈ΛThisisthementionedcompatibilitycondition.Itensuresthesolvabilityofthediscretesystem(6.24).InthecaseoftheDonalddiagram,weobviouslyhavek∇uh,∇vh0=0.Sincetheproofof(6.25)doesnotdependontheparticulartypeofthecontrolvolumes,thepropertyofdiscreteglobalconservativityinthesenseof(6.26)issatisfiedfortheDonalddiagram,too.InverseMonotonicityTheso-calledinversemonotonicityisafurtherimportantpropertyoftheboundaryvalueproblem(6.5)thatisinheritedbythefinitevolumedis-cretizationwithoutanyadditionalrestrictiveassumptions.Namely,itiswellknownthatunderappropriateassumptionsonthecoefficients,thesolutionuisnonnegativeifthe(continuous)right-handsidefin(6.5)isnonnegativeinΩ.Wewilldemonstratethatthisremainstruefortheapproximativesolu-tionuh.Onlyatthisplaceistheproperty(P3)oftheweightingfunctionRused;theprecedingresultsarealsovalidforthesimplecaseR(z)≡1.2ThereisacloserelationtothemaximumprinciplesinvestigatedinSec-tions1.4and3.9.However,theresultgivenhereisweaker,andtheproofisbasedonadifferenttechnique.Theorem6.19LettheassumptionsofTheorem6.15besatisfied,butRin(6.12)hastosatisfy(P1)–(P3).Further,supposethatf∈C(Ω)andf(x)≥0forallx∈Ω.Moreover,inthecaseoftheDonalddiagram,onlytheweightingfunctionR(z)=1[sign(z)+1]ispermitted.2Thenuh(x)≥0forallx∈Ω.Proof:WestartwiththecaseoftheVoronoidiagram.Letuhbethesolutionof(6.11)withf(x)≥0forallx∈Ω.Thenwehavethefollowingadditivedecompositionofuh:+−+uh=uh−uh,whereuh:=max{0,uh}.+−Ingeneral,uh,uhdonotbelongtoVh.SoweinterpolatetheminVhandsetin(6.11)v:=I(u−),whereI:CΩ¯→Vistheinterpolationhhhhh 6.2.FiniteVolumeMethodonTriangularGrids281operator(3.71).Itfollowsthat+−−−0≤f,vh0,h=ah(uh,vh)=ahIh(uh),Ih(uh)−ahIh(uh),Ih(uh).ByTheorem6.15,wehave2kI(u−)≤aI(u−),I(u−)≤aI(u+),I(u−).hh1hhhhhhhhhh+−IfwewereabletoshowthatahIh(uh),Ih(uh)≤0,thenthetheoremwouldbeproven,becausethisrelationimpliesI(u−)=0,andfromthishh1−+weimmediatelygetuh=0,andsouh=uh≥0.+−Sinceuiui=0foralli∈Λ,itfollowsfrom(6.19)intheproofofLemma6.12that+−+−bhIh(uh),Ih(uh)=(1−rij)ujuiγijmij.(6.27)i∈Λj∈Λi+−Furthermore,obviouslydhIh(uh),Ih(uh)=0holds.Thus56+−µij++−ahIh(uh),Ih(uh)=−uj+γij(1−rij)ujuimijdiji∈Λj∈Λi56µijγijdij+−=−1−(1−rij)ujuimij.dijµiji∈Λj∈ΛiDueto1−[1−R(z)]z≥0forallz∈R(cf.property(P3)in(6.13))and+−ujui≥0,itfollowsthat+−ahIh(uh),Ih(uh)≤0.SoitremainstoinvestigatethecaseoftheDonalddiagram.ThefunctionR(z)=1[sign(z)+1]hastheproperty21[1−R(z)]z=[1−sign(z)]z≤0forallz∈R,2thatis(cf.(6.27)),+−bhIh(uh),Ih(uh)≤0.+−Takinguiui=0intoconsideration,weget34+−+−ahIh(uh),Ih(uh)≤k∇Ih(uh),∇Ih(uh)0+−=kujui∇ϕj,∇ϕi0.i∈Λj∈ΛiNowLemma3.47impliesthatkaI(u+),I(u−)≤−u+u−cotαK+cotαK,hhhhh2jiijiji∈Λj∈ΛiwhereKandK
areapairoftrianglessharingacommonedgewithverticesai,aj. 2826.FiniteVolumeMethodSincealltrianglesarenonobtuse,wehavecotαK≥0,cotαK≥0,andijijhence+−ahIh(uh),Ih(uh)≤0.
Exercises6.2SupposethatthedomainΩ⊂R2canbetriangulatedbymeansofequilateraltriangleswithedgelengthh>0inanadmissibleway.(a)GivetheshapeofthecontroldomainsinthecaseoftheVoronoiandtheDonalddiagrams.(b)Usingthecontroldomainsfromsubproblem(a),discretizethePoissonequationwithhomogeneousDirichletboundaryconditionsbymeansofthefinitevolumemethod.6.3FormulateanexistenceresultfortheweaksolutioninH1(Ω)ofthe0boundaryvalueproblem(6.5)similartoTheorem3.12.Inparticular,whatformwillcondition(3.17)take?6.4VerifyRemark6.7;i.e.,showthat·,·possessesthepropertiesof0,hascalarproductonVh.6.5ProveRemark6.16indetail.6.6Verifyordisprovetheproperties(P1)–(P3)forthethreeweightingfunctionsgivenbefore(6.13)andforR≡1.26.7LetKbeanonobtusetrianglewiththeverticesa1,a2,a3.ThelengthofthesegmentsΓK:=Γ∩KisdenotedbymK,anddisthelengthijijijijoftheedgeconnectingawitha.Finally,αKistheinteriorangleofKijijoppositethatedge.Demonstratethefollowingrelation:2mK=dcotαK.ijijij6.8(a)Formulateproblem(6.11)intermsofanalgebraicsystemoftype(1.31).(b)ShowthatfortheresultingmatrixA∈RM1,M1,whereMistheh1numberofelementsoftheindexsetΛ,thefollowingrelationisvalid:AT1≥0.Here,asinSection1.4,0,respectively1,denotesavectorhofdimensionM1whosecomponentsareallequalto0,respectively1.(Thisisnothingotherthantheproperty(1.32)(3)(i)exceptforthetransposeofAh.) 7DiscretizationMethodsforParabolicInitialBoundaryValueProblems7.1ProblemSettingandSolutionConceptInthissectioninitialboundaryvalueproblemsforthelinearcaseofthedifferentialequation(0.33)areconsidered.Wechoosetheform(3.12)to-getherwiththeboundaryconditions(3.18)–(3.20),whichhavealreadybeendiscussedinSection0.5.InSection3.2conditionshavebeendevelopedtoensureauniqueweaksolutionofthestationaryboundaryvalueproblem.IncontrasttoChapter3,theheterogeneitiesarenowallowedalsotodependontimet,butforthesakeofsimplicitywedonotdosoforthecoefficientsinthedifferentialequationsandtheboundaryconditions,whichcoversmostoftheapplications,forexamplefromChapter0.Alsoforthesakeofsimplicity,wetakethecoefficientinfrontofthetimederivativetobeconstantandthus1byaproperscaling.FromtimetotimewewillrestrictattentiontohomogeneousDirichletboundaryconditionsforfurthereaseofexposition.Thustheproblemreadsasfollows:ThedomainΩisassumedtobeaboundedLipschitzdomainandwesupposethatΓ1,Γ2,Γ3formadisjointdecompositionoftheboundary∂Ω(cf.(0.39)):∂Ω=Γ1∪Γ2∪Γ3,whereΓ3isaclosedsubsetoftheboundary.Inthespace-timecylinderQT=Ω×(0,T),T>0,anditsboundaryST=∂Ω×(0,T)therearegivenfunctionsf:QT→R,g:ST→R,g(x,t)=gi(x,t)forx∈Γi,i=1,2,3,andu0:Ω→R.Theproblemisto 2847.DiscretizationofParabolicProblemsfindafunctionu:QT→Rsuchthat∂u+Lu=finQT,∂tRu=gonST,(7.1)u=u0onΩ×{0},whereLvdenotesthedifferentialexpressionforsomefunctionv:Ω→R,(Lv)(x):=−∇·(K(x)∇v(x))+c(x)·∇v(x)+r(x)v(x)(7.2)withsufficientlysmooth,time-independentcoefficientsd,ddK:Ω→R,c:Ω→R,r:Ω→R.TheboundaryconditionisexpressedbytheshorthandnotationRu=g,whichmeans,forafunctionα:Γ2→Ron∂Ω,•Neumannboundarycondition(cf.(0.41)or(0.36))K∇u·ν=∂νKu=g1onΓ1×(0,T),(7.3)•mixedboundarycondition(cf.(0.37))K∇u·ν+αu=∂νKu+αu=g2onΓ2×(0,T),(7.4)•Dirichletboundarycondition(cf.(0.38))u=g3onΓ3×(0,T).(7.5)ThusthestationaryboundaryproblemconsideredsofarreadsLu(x)=f(x)forx∈Ω,(7.6)Ru(x)=g(x)forx∈∂Ω.Itistobeexpectedthatbothfortheanalysisandthediscretizationtherearestronglinksbetween(7.6)and(7.1).Theformulation(7.1)inparticularincludestheheatequation(cf.(0.20))∂u−∇·(K∇u)=finQT,(7.7)∂torforconstantscalarcoefficientsintheform(cf.(0.19))∂u−∆u=finQT(7.8)∂twithappropriateinitialandboundaryconditions.AgainasinChapter1,oneofthesimplestcaseswillbe,fortwospacedimensions(d=2),thecaseofarectangleΩ=(0,a)×(0,b)oreventhecased=1(withΩ=(0,a)),forwhich(7.8)furtherreducesto∂u∂2−u=0inQT=(0,a)×(0,T).(7.9)∂t∂x2Forproblem(7.1),thefollowingtypicalanalyticalquestionsarise: 7.1.ProblemSettingandSolutionConcept285•existenceof(classical)solutions,•propertiesofthe(classical)solutions,•weakerconceptsofthesolution.Asinthecaseofellipticboundaryvalueproblems,thetheoryofclassicalsolutionsrequirescomparativelystrongassumptionsonthedataoftheinitial-boundaryvalueproblem.Inparticular,alongtheedge∂Ω×{0}ofthespace-timecylinderinitialandboundaryconditionsmeet,sothatadditionalcompatibilityconditionshavetobetakenintoaccount.RepresentationofSolutionsinaSpecialCaseToenhancethefamiliaritywiththeproblemandforfurthercomparisonwebrieflysketchamethod,namedseparationofvariables,bywhichclosed-formsolutionsintheformofinfiniteseriescanbeobtainedforspecialcases.Alsointhesecases,therepresentationsarenotmeanttobeanumericalmethod(byitsevaluation),butonlyserveasatheoreticaltool.Westartwiththecaseofhomogeneousdata,i.e.,f=0,gi=0(i=1,2,3),sothattheprocessisdeterminedonlybytheinitialdatau0.Weassumeasolutionof(7.1)tohavetheformu(x,t)=v(t)w(x)withv(t)=0,w(x)=0.Thisleadstov
(t)−Lw(x)=,x∈Ω,t∈(0,T).(7.10)v(t)w(x)Therefore,theexpressionsin(7.10)mustbeconstant,forexample,equalto−λforλ∈R.Therefore,
v(t)=−λv(t),t∈(0,T),(7.11)whichfortheinitialconditionsv(0)=1hasthesolution−λtv(t)=e.Furthermore,whastosatisfyLw(x)=λw(x),x∈Ω,(7.12)Rw(x)=0,x∈∂Ω.Suchafunctionw:Ω¯→R,w=0,iscalledaneigenfunctionfortheeigen-valueλoftheboundaryvalueproblem(7.6).If(wi,λi),i=1,...,N,areeigenfunctions/valuesfor(7.6),thenbecauseofthesuperpositionprinciple,thefunctionNu(x,t):=ce−λitw(x)(7.13)iii=1 2867.DiscretizationofParabolicProblemsisasolutionofthehomogeneousinitial-boundaryvalueproblemfortheinitialvalueNu0(x):=ciwi(x),(7.14)i=1wheretheci∈Rarearbitrary.Ifthereareinfinitelymanyeigenfunc-tions/values(wi,λi)andifthesumsin(7.13)and(7.14)convergeinsuchawaythatalsotheinfiniteseriespossessesthederivativesappearingin(7.6),thenalso∞u(x,t)=ce−λitw(x)(7.15)iii=1isasolutionto∞u0(x)=ciwi(x).(7.16)i=1Foraninhomogeneousright-handsideoftheformNf(x,t)=fi(t)wi(x)(7.17)i=1thesolutionrepresentationcanbeextendedto(variationofconstantsformula)NN
tu(x,t):=ce−λitw(x)+f(s)e−λi(t−s)dsw(x),(7.18)iiiii=1i=10andatleastformallythesumcanbereplacedbytheinfiniteseries.Toverify(7.18)itsufficestoconsiderthecaseu0=0,forwhichwehaveNNt(∂u)(x,t)=f(t)w(x)−f(s)e−λi(t−s)dsλw(x)tiiiiii=1i=10Nt(7.19)=f(x,t)−Lf(s)e−λi(t−s)dsw(x)iii=10=f(x,t)−L(u)(x,t).Fromthesesolutionrepresentationswecanconcludethatinitialdata(andthusalsoperturbancescontainedinit)andalsotheinfluenceoftheright-handsideactonlyexponentiallydampedifalleigenvaluesarepositive.Ford=1,Ω=(0,a)andDirichletboundaryconditionswehavetheeigenfunctionsνπw(x)=sinνx,ν∈N,(7.20)a 7.1.ProblemSettingandSolutionConcept287fortheeigenvaluesνπ2λν=.(7.21)aIftheinitialdatau0hastherepresentation∞πu0(x)=cisinνx,(7.22)aν=1thenforexampleforf=0the(formal)solutionreads∞πu(x,t)=ce−λνtsinνx.(7.23)iaν=1Theeigenfunctionswνareorthogonalwithrespecttothescalarproduct·,·inL2(Ω),sincetheysatisfy0$;ππ<0forν=µ,sinν·,sinµ·=(7.24)aa0aforν=µ,2whichcanbycheckedbymeansofwell-knownidentitiesforthetrigono-metricfunctions.Therefore(seebelow(7.57)),u,wν00ci=,(7.25)wν,wν0whichiscalledtheFouriercoefficientintheFourierexpansionofu0.Ofcourse,the(wν,λν)dependontheboundaryconditions.ForNeumannboundaryconditionsinx=0andx=awehavewν(x)=cosνπx,ν=0,1,...,a2(7.26)λν=νπ,ν=0,1,....aTheoccurrenceofw0=1,λ0=0reflectsthenontrivialsolvabilityofthepureNeumannproblem(whichthereforeisexcludedbytheconditionsofTheorem3.15).ForLu=−∆uandΩ=(0,a)×(0,b),eigenfunctionsandeigenvaluescanbederivedfromtheone-dimensionalcasebecauseof−∆(vν(x)˜vµ(y))=−vν

(x)˜vµ(y)−vν(x)˜vµ

(y)=(λν+λ˜µ)vν(x)˜vµ(y).Therefore,forΩ=(0,a)×(0,b)onehastochoosetheeigenfunctions/values(vν,λν)(inx,on(0,a))fortherequiredboundaryconditionsatx=0andx=a,and(˜vµ,λ˜µ)(iny,on(0,b))fortherequiredboundaryconditionsaty=0,y=b.ForDirichletboundaryconditionseverywherethisleadstoνµππw(x,y)=sinνxsinµx(7.27)ab 2887.DiscretizationofParabolicProblemsfortheeigenvaluesνπ2µπ2λνµ=+abπ2π2νµ(i.e.,thesmallesteigenvalueis+andλ→∞forν→∞orabµ→∞).Asafurtherconcludingexamplewenotethecasex=0orx=a:u(x,y)=0fory∈[0,b],y=0:∇u·ν(x,y)=−∂2u(x,y)=0forx∈(0,a),y=b:∇u·ν(x,y)=∂2u(x,y)=0forx∈(0,a).Eigenfunctions:νµππw(x,y)=sinνxcosµy,(7.28)abν=1,2,...,µ=0,1,2,....Eigenvalues:π2π2λνµ=ν+µ.abASketchoftheTheoryofWeakSolutionsAsinthestudyoftheellipticboundaryvalueproblems(3.12),(3.18)–(3.20),forequation(7.1)aweakformulationcanbegiventhatreducestherequirementswithrespecttothedifferentiabilitypropertiesofthesolution.Theideaistotreattimeandspacevariablesinadifferentway:(1)•Forfixedt∈(0,T),thefunctionx→u(x,t)isinterpretedasaparameter-dependentelementu(t)ofsomespaceVwhoseelementsarefunctionsofx∈Ω.Anobviouschoiceis(seeSubsection3.2.1,(I))thespace1V={v∈H(Ω):v=0onΓ3}.•Inanextstep,thatis,forvaryingt,afunctiont→u(t)resultswithvaluesinthe(function)spaceV.(2)InadditiontoV,afurtherspaceH=L2(Ω)occurs,fromwhichtheinitialvalueu0istakenandwhichcontainsVasadensesubspace.AsubspaceViscalleddenseinHiftheclosureofVwithrespecttothenormonHcoincideswithH.(3)Thetimederivativeisunderstoodinageneralizedsense;see(7.29).(4)Thegeneralizedsolutiont→u(t)issoughtasanelementofafunctionspace,theelementsofwhichare“function-valued”(cf.(1)).Definition7.1LetXdenoteoneofthespacesHorV(inparticular,thismeansthattheelementsofXarefunctionsonΩ⊂Rd). 7.1.ProblemSettingandSolutionConcept289(i)ThespaceCl([0,T],X),l∈N,consistsofallcontinuousfunctions0v:[0,T]→Xthathavecontinuousderivativesuptotheorderlon[0,T]withthenormlsupv(i)(t).Xt∈(0,T)i=0Forthesakeofsimplicity,thenotationC([0,T],X):=C0([0,T],X)isused.(ii)ThespaceLp((0,T),X)with1≤p≤∞consistsofallfunctionson(0,T)×Ωwiththefollowingproperties:v(t,·)∈Xforanyt∈(0,T),F∈Lp(0,T)withF(t):=v(t,·).XFurthermore,vLp((0,T),X):=FLp(0,T).Remark7.2f∈L2(Q)⇒f∈L2((0,T),H).TProof:Basically,theproofisaconsequenceofFubini’stheorem(see[1]).
Concerningtheinterpretationofthetimederivativeandoftheweakformulation,acomprehensivetreatmentispossibleonlywithintheframe-workofthetheoryofdistributions;thusadetailedexplanationisbeyondthescopeofthisbook.Ashortbutmathematicallyrigorousintroductioncanbefoundinthebook[39,Chapter23].Thebasicideaconsistsinthefollowingdefinition:Afunctionu∈L2((0,T),V)issaidtohaveaweakderivativewifthefollowingholds:
T
Tu(t)Ψ
(t)dt=−w(t)Ψ(t)dtforallΨ∈C∞(0,T).(7.29)000Usually,thisderivativewisdenotedbyduoru
.dtRemark7.3Theintegralsoccurringabovearetobeunderstoodasso-calledBochnerintegralsandareextensionsoftheLebesgueintegraltofunction-valuedmappings.Therefore,equation(7.29)isanequalityoffunctions.Beforewegiveaweakformulationof(7.1),thefollowingnotionisworthrecalling:
u,v:=uvdxforu,v∈H,(7.30)0Ω

a(u,v):=[K∇u·∇v+(c·∇u+ru)v]dx+αuvdσ,u,v∈V.(7.31)ΩΓ2 2907.DiscretizationofParabolicProblemsLetu∈H,f∈L2((0,T),H),andincaseofDirichletconditionswe0restrictourselvestothehomogeneouscase.Anelementu∈L2((0,T),V)iscalledaweaksolutionof(7.1)ifithasaweakderivativedu=u
∈L2((0,T),H)andthefollowingholdsdt=>du(t),v+a(u(t),v)=f(t),v0+g1(·,t)vdσdt0Γ1+g2(·,t)vdσ(7.32)Γ2forallv∈Vandt∈(0,T),u(0)=u0.Duetou∈L2((0,T),V)andu
∈L2((0,T),H),wealsohaveu∈C([0,T],H)(see[12,p.287]),sothattheinitialconditionismeaningfulintheclassicalsense.Inwhatfollows,thebilinearformaisassumedtobecontinuousonV×V(see(3.2))andV-elliptic(see(3.3)).Thelattermeansthatthereexistsanumberα>0suchthata(v,v)≥αv2forallv∈V.VLemma7.4LetabeaV-elliptic,continuousbilinearform,u0∈H,andf∈C([0,T],H),andsupposetheconsideredboundaryconditionsareho-mogeneous.Then,forthesolutionu(t)of(7.32)thefollowingestimateholds:
tu(t)≤ue−αt+f(s)e−α(t−s)dsforallt∈(0,T).00000Proof:Thefollowingequationsarevalidalmosteverywherein(0,T).Settingv=u(t),(7.32)readsas
u(t),u(t)+a(u(t),u(t))=f(t),u(t).00Usingtherelation
1d1d2du(t),u(t)0=u(t),u(t)0=u(t)0=u(t)0u(t)02dt2dtdtandtheV-ellipticity,itfollowsthatd2u(t)0u(t)0+αu(t)V≤f(t),u(t)0.dtNowthesimpleinequalityu(t)0≤u(t)VandtheCauchy–Schwarzinequalityf(t),u(t)0≤f(t)0u(t)0 7.1.ProblemSettingandSolutionConcept291yield,afterdivisionbyu(t)0,theestimatedu(t)0+αu(t)0≤f(t)0.dtMultiplyingthisrelationbyeαt,therelationdαtαtdαt(eu(t)0)=eu(t)0+αeu(t)0dtdtleadstodαtαt(eu(t)0)≤ef(t)0.dtTheintegrationover(0,t)resultsin
teαtu(t)−u(0)≤eαsf(s)ds0000forallt∈(0,T).Multiplyingthisbye−αtandtakingintoconsiderationtheinitialcondition,wegettheassertedrelation
tu(t)≤ue−αt+f(s)e−α(t−s)ds.00000
Asaconsequenceofthislemma,theuniquenessofthesolutionof(7.32)isobtained.Corollary7.5LetabeaV-elliptic,continuousbilinearform.Thenthereexistsatmostonesolutionof(7.32).Proof:Supposetherearetwodifferentsolutionsu1(t),u2(t)∈V.Thenthedifferencev(t):=u1(t)−u2(t)solvesahomogeneousproblemofthetype(7.32)(i.e.,withf=0,u0=0).Lemma7.4immediatelyimpliesv(t)0=0in[0,T);thatis,u1(t)=u2(t)forallt∈[0,T).
ThereisacloserelationbetweenLemma7.4andsolutionrepresentationssuchas(7.18)(withthesumbeinginfinite).Theeigenvalueproblem(7.12)isdefinedasfollowsinitsvariationalform(seealsotheendofSection2.2):Definition7.6Anumberλ∈Riscalledaneigenvaluefortheeigenvectorw∈V,w=0,ifa(w,v)=λw,v0forallv∈V.AssumethatadditionallytoourassumptionsthebilinearformissymmetricandtheembeddingofVintoHiscompact(see[26]),whichisthecasehere.Thenthereareenougheigenvectorsinthesensethatasequence(wi,λi), 2927.DiscretizationofParabolicProblems0<λ1≤λ2≤...,existssuchthatthewiareorthonormalwithrespectto·,·0andeveryv∈Vhasauniquerepresentation(inH)as∞v=ciwi.(7.33)i=1Asin(7.25)theFouriercoefficientsciaregivenbyci=v,wi0.(7.34)Infact,(7.33)givesarigorousframeworktothespecificconsiderationsin(7.16)andsubsequentformulas.From(7.33)and(7.34)weconcludeParseval’sidentity∞v2=|v,w|2.(7.35)0i0i=1−1/2Furthermore,thesequencevi:=λiwiisorthogonalwithrespecttoa(·,·),andarepresentationcorrespondingto(7.33),(7.34)holdssuchthat∞∞∞a(v,v)=|a(v,v)|2=λ−1|a(v,w)|2=λ|v,w|2.(7.36)iiiii0i=1i=1i=1From(7.35)and(7.36)weseethattheellipticityconstantcanbeinter-pretedasthesmallesteigenvalueλ.Infact,thesolutionrepresentation(7.18)(withthesumbeinginfiniteinH)alsoholdstrueundertheas-sumptionsmentionedandalsoleadstotheestimateofLemma7.4.Butnotethattheprooftheredoesnotrequiresymmetryofthebilinearform.Exercises7.1Considertheinitial-boundaryvalueproblemut−uxx=0in(0,∞)×(0,∞),u(0,t)=h(t),t∈(0,∞),u(x,0)=0,x∈(0,∞),whereh:(0,∞)→Risadifferentiablefunction,thederivativeofwhichhasatmostexponentialgrowth.(a)Showthatthefunction?
∞22−s2/2xu(x,t)=eht−dsπ√2s2x/2tisasolution.(b)Isutboundedinthedomainofdefinition?Ifnot,giveconditionsonhthatguaranteetheboundednessofut. 7.2.SemidiscretizationbytheVerticalMethodofLines2937.2Considertheinitial-boundaryvalueprobleminonespacedimensionut−uxx=0in(0,π)×(0,∞),u(0,t)=u(π,t)=0,t∈(0,∞),u(x,0)=u0(x),x∈(0,π).(a)Solveitbymeansofthemethodofseparation.(b)Givearepresentationforut(t)0.(c)Considertheparticularinitialconditionu0(x)=π−xandinvestigate,usingtheresultfromsubproblem(b),theasymptoticbehaviourofut(t)0neart=0.7.3LetthedomainΩ⊂RdbeboundedbyasufficientlysmoothboundaryandsetV:=H1(Ω),H:=L2(Ω).Furthermore,a:V×V→Risa0continuous,V-elliptic,symmetricbilinearformandu0∈H.Provebyusingtheso-calledenergymethod(cf.theproofofLemma7.4)thefollowingaprioriestimateforthesolutionuoftheinitialboundaryvalueproblemut(t),v0+a(u(t),v)=0forallv∈V,t∈(0,T),u(0)=u0.
t
t(a)αtu(t)2+2su(s)2ds≤Mu(s)2ds.1t01?00M1(b)ut(t)0≤u00.2αtHereMandαdenotethecorrespondingconstantsinthecontinuityandellipticityconditions,respectively.7.2SemidiscretizationbytheVerticalMethodofLinesForsolvingparabolicequationsnumerically,awidevarietyofmethodsexists.Themostimportantclassesofthesemethodsarethefollowing:•Fulldiscretizations:–Applicationoffinitedifferencemethodstotheclassicalinitialboundaryvalueproblem(asoftheform(7.1)).–Applicationofso-calledspace-timefiniteelementmethodstoavariationalformulationthatincludesthetimevariable,too.•Semidiscretizations:–Theverticalmethodoflines:Herethediscretizationstartswithrespecttothespatialvariable(s)(e.g.,bymeansofthefinitedif- 2947.DiscretizationofParabolicProblemsferencemethod,thefiniteelementmethod,orthefinitevolumemethod).–Thehorizontalmethodoflines(Rothe’smethod):Herethediscretizationstartswithrespecttothetimevariable.Asthenameindicates,asemidiscretizationhastobefollowedbyafurtherdiscretizationsteptoobtainafulldiscretization,whichmaybeoneoftheabove-mentionedornot.Theideabehindsemidiscretizationmethodsistohaveintermediateproblemsthatareofawell-knownstructure.Inthecaseoftheverticalmethodoflines,asystemofordinarydifferentialequationsarisesforthesolutionofwhichappropriatesolversareoftenavailable.Rothe’smethodgeneratesasequenceofellipticboundaryvalueproblemsforwhichefficientsolutionmethodsareknown,too.Theattributes“vertical”and“horizontal”ofthesemidiscretizationsaremotivatedbythegraphicalrepresentationofthedomainofdefinitionoftheunknownfunctionu=u(x,t)inonespacedimension(i.e.,d=1),namely,assigningtheabscissa(horizontalaxis)ofthecoordinatesystemtothevariablexandtheordinate(verticalaxis)tothevariablet,sothatthespatialdiscretizationyieldsproblemsthataresettedalongverticallines.Inwhatfollows,theverticalmethodoflineswillbeconsideredinmoredetail.Inthefollowing,andsimilarlyinthefollowingsections,wewillde-veloptheanalogous(semi)discretizationapproachesforthefinitedifferencemethod,thefiniteelementmethod,andthefinitevolumemethod.Thiswillallowustoanalyzethesemethodsinauniformway,asfarasonlytheemerging(matrix)structureofthediscreteproblemswillplayarole.Ontheotherhand,differenttechniquesofanalysisasinChapters1,3and6willfurtherelucidateadvantagesanddisadvantagesofthemethods.Read-erswhoareinterestedonlyinaspecificapproachmayskipsomeofthefollowingsubsections.TheVerticalMethodofLinesfortheFiniteDifferenceMethodAsafirstexamplewestartwiththeheatequation(7.8)withDirichletboundaryconditionsonarectangleΩ=(0,a)×(0,b).AsinSec-tion1.2weapplythefive-pointstencildiscretizationsatthegridpointsx∈Ωh(accordingto(1.5))foreveryfixedt∈[0,T].Thisleadstotheapproximation1∂tuij(t)+2−ui,j−1(t)−ui−1,j(t)+4uij(t)−ui+1,j(t)−ui,j+1(t)h=fij(t),i=1,...,l−1,j=1,...,m−1,t∈(0,T),(7.37)uij(t)=gij(t),i∈{0,l},j=0,...,m,(7.38)j∈{0,m},i=0,...,l. 7.2.SemidiscretizationbyVerticalMethodofLines295Hereweusefij(t):=f(ih,jh,t),(7.39)gij(t):=g(ih,jh,t),andtheindex3intheboundaryconditionisomitted.Additionally,theinitialcondition(atthegridpoints)willbeprescribed,thatis,uij(0)=u0(ih,jh),(ih,jh)∈Ωh.(7.40)Thesystem(7.37),(7.38),(7.40)isasystemof(linear)ordinarydifferentialequations(inthe“index”(i,j)).If,asinSection1.2,wefixanorderingofthegridpoints,thesystemtakestheformduh(t)+Ahuh(t)=qh(t),t∈(0,T),dt(7.41)uh(0)=u0,withAh,qhasin(1.10),(1.11)(butnowqh=qh(t)becauseofthet-dependenceoffandg).Theunknownisthefunctionu:[0,T]→RM1,(7.42)hwhichmeansthattheDirichletboundaryconditionsareeliminatedasinSection1.2.ForasimplificationofthenotationweuseinthefollowingMinsteadofM1,whichalsoincludestheeliminateddegreesoffreedom.OnlyinSections7.5and7.6willwereturntotheoriginalnotation.Moregenerally,ifweconsiderafinitedifferenceapproximation,whichappliedtothestationaryproblem(7.6)willleadtothesystemofequationsAhuh=qh,withu∈RM,thenthesamemethodappliedto(7.1)foreveryfixedht∈(0,T)leadsto(7.41).Inparticular,thesystem(7.41)hasauniquesolutionduetothetheoremofPicard–Lindel¨of(cf.[26]).TheVerticalMethodofLinesfortheFiniteElementMethodWeproceedasforthefinitedifferencemethodbynowapplyingthefiniteelementmethodto(7.1)initsweakformulation(7.32)foreveryfixedt∈(0,T),usingtheabbrevation

b(t,v):=f(t),v0+g1(·,t)vdσ+g2(·,t)vdσ.(7.43)Γ1Γ2SoletVh⊂Vdenoteafinite-dimensionalsubspacewithdimVh=M=M(h)andletu0h∈Vhbesomeapproximationtou0.Thenthesemidiscreteproblemreadsasfollows: 2967.DiscretizationofParabolicProblemsFindu∈L2((0,T),V)withu
∈L2((0,T),H),u(0)=uandhhhh0h=>duh(t),vh+a(uh(t),vh)=b(t,vh)forallvh∈Vh,t∈(0,T).(7.44)dt0Togainamorespecificformof(7.44),againwerepresenttheunknownuh(t)byitsdegreesoffreedom:MMLet{ϕi}i=1beabasisofVh,uh(t)=i=1ξi(t)ϕiandu0h=Mi=1ξ0iϕi.Thenforanyt∈(0,T),thediscretevariationalequality(7.44)isequivalenttoMdξ(t)Mjϕj,ϕi0+a(ϕj,ϕi)ξj(t)=b(t,ϕi)foralli∈{1,...,M}.dtj=1j=1DenotingbyAˆh:=(a(ϕj,ϕi))ijthestiffnessmatrix,byBh:=ϕj,ϕi0ijthemassmatrix,andbyβh(t):=(b(t,ϕi))i,respectivelyξ0h:=(ξ0i)i,thevectorsoftheright-handsideandoftheinitialvalue,weobtainforξh(t):=(ξi(t))ithefollowingsystemoflinearordinarydifferentialequationswithconstantcoefficients:dBhξh(t)+Aˆhξh(t)=βh(t),t∈(0,T),(7.45)dtξh(0)=ξ0h.SincethematrixBhissymmetricandpositivedefinite,itcanbefactored(e.g.,bymeansofCholesky’sdecomposition)asB=ETE.Introducinghhhthenewvariableuh:=Ehξh(tomaintainthepossibledefinitenessofAh),theabovesystem(7.45)canbewrittenasfollows:duh(t)+Ahuh(t)=qh(t),t∈(0,T),(7.46)dtuh(0)=uh0,whereA:=E−TAˆE−1isanRM-ellipticmatrixandq:=E−Tβ,hhhhhhhuh0:=Ehξ0h.Thusagainthediscretizationleadsustoasystem(7.41).Remark7.7BymeansofthesameargumentsasintheproofofLemma7.4,anestimateofuh(t)0canbederived.TheVerticalMethodofLinesfortheFiniteVolumeMethodBasedonthefinitevolumemethodsintroducedinChapter6,inthissub-sectionafinitevolumesemidiscretizationisgivenfortheproblem(7.1)initsweakformulation(7.32)foreveryfixedt∈(0,T)inthespecialcaseΓ3=∂ΩandofhomogeneousDirichletboundaryconditions.AsinChapter6,theonlyessentialdifferencetoproblem(7.1)isthatherethe 7.2.SemidiscretizationbyVerticalMethodofLines297differentialexpressionLisindivergenceform,i.e.,Lu:=−∇·(K∇u−cu)+ru=f,wherethedataK,c,r,andfareasin(7.2).Correspondingly,thebilinearformaintheweakformulation(7.32)istobereplacedby
a(u,v)=[(K∇u−cu)·∇v+ruv]dx.(7.47)ΩInordertoobtainafinitevolumesemidiscretizationoftheproblem(7.1)indivergenceform,andof(7.32)withthemodification(7.47),werecallthewaythatitwasdoneintheellipticsituation.Namely,comparingtheweakformulationoftheellipticproblem(seeDefinition2.2)withthefinitevolumemethodinthediscretevariationalformulation(6.11),weseethatthebilinearformaandthelinearformb(·):=f,·havebeenreplacedby0certaindiscreteformsahandf,·0,h,respectively.Thisformalprocedurecanbeappliedtotheweakformulation(7.32)oftheparabolicproblem,too.SoletVh⊂Vdenoteafinite-dimensionalsubspaceasintroducedinSec-tion6.2withdimVh=M=M(h)andletu0h∈Vhbesomeapproximationtou0.Then,thesemidiscretefinitevolumemethodreadsasfollows:Findu∈L2((0,T),V)withu
∈L2((0,T),H),u(0)=uandhhhh0h;0independentofh.Then,takingvh=uh(t)in(7.48),weget;0.1−q1 7.2.SemidiscretizationbyVerticalMethodofLines303∞µqlFinally,weconcludetheestimatebecauseofq=byµ=l1−qqlqm12≤C1C2+C3.1−q11−q2Therefore,thiserrorcontributionfort≥t(forafixedt>0)approaches0forl→∞andm→∞.Thelargertis,themorethiserrortermwill2decrease.Becauseof,forexample,l=a/handthusql=exp−πt1,1ahthedecayinhisexponentialandthusmuchstrongerthanatermlikeO(h2).Therefore,wehavetocomparethetermsinthesumonlyforν=1,...,l−1,µ=1,...,m−1,i.e.,theerrorintheFouriercoefficientandintheeigenvalue:νµνµνµνµνµ−λth−λth−λth−λt−λtcνµe−cνµeh=cνµ−cνµe+cνµe−eh.Notethatchcanbeperceivedasanapproximationofcbythetrape-νµνµzoidalsumwithstepsizehineachspatialdirection(see,e.g.,[30],p.129),sincetheintegrandinthedefinitionofcνµvanishesforx=0orx=aandy∈[0,b],andy=0ory=bandx∈[0,a].Thuswehaveforu∈C2(Ω)¯,0h2|cνµ−cνµ|=O(h).Becauseof−λνµt−λνµt−λνµt−(λνµ−λνµ)te−eh=e1−eh,and|λνµ−λνµ|=O(h2)(see(7.61)),alsothistermisoforderO(h2)andhwillbedampedexponentially(dependingontandthesizeofthesmallesteigenvalueλνµ).Summarizing,weexpect2O(h)tobethedominatingerrorterminthediscretemaximumnorm·∞atthegridpoints(cf.definition(1.17)),whichwillalsobedampedexponentiallyforincreasingt.NotethatwehavegivenonlyheuristicargumentsandthattheconsiderationscannotbetransferredtotheNeumanncase,wheretheeigenvalueλ=0appears.Wenowturntothefiniteelementmethod.OrderofConvergenceEstimatesfortheFiniteElementMethodWewillinvestigatethefiniteelementmethodonamoreabstractlevelasintheprevioussubsection,butwewillachievearesult(indifferentnorms)ofsimilarcharacter.AsworkedoutattheendofSection7.1,thereisastrongrelationbetweentheV-ellipticityofthebilinearformawiththeparameterαandapositivelowerboundoftheeigenvalues.HerewerelyontheresultsalreadyachievedinSection2.3andSection3.4forthestationarycase.Forthat,weintroducetheso-calledellipticprojectionofthesolutionu(t)of(7.32)asaveryimportanttoolintheproof. 3047.DiscretizationofParabolicProblemsDefinition7.8ForaV-elliptic,continuousbilinearforma:V×V→R,theelliptic,orRitz,projectionRh:V→Vhisdefinedbyv→Rhv⇐⇒a(Rhv−v,vh)=0forallvh∈Vh.Theorem7.9UndertheassumptionsofDefinition7.8:(i)Rh:V→Vhislinearandcontinuous.(ii)Rhyieldsquasi-optimalapproximations;thatis,Mv−RhvV≤infv−vhV,αvh∈VhwhereMandαaretheLipschitzandellipticityconstantsaccordingto(2.42)and(2.43).Proof:ThelinearityofRhisobvious.TheremainingstatementsimmediatelyfollowfromLemma2.16andTheorem2.17;seeExercise7.6.
Makinguseoftheellipticprojection,weareabletoprovethefollowingresult.Theorem7.10SupposeaisaV-elliptic,continuousbilinearform,f∈C([0,T],H),u0∈V,andu0h∈Vh.Thenifu(t)issufficientlysmooth,u(t)−u(t)≤u−Rue−αt+(I−R)u(t)h00hh00h0
t+(I−R)u
(s)e−α(t−s)ds.h00Proof:First,theerrorisdecomposedasfollows:uh(t)−u(t)=uh(t)−Rhu(t)+Rhu(t)−u(t)=:θ(t)+(t).Wetakev=vh∈Vhin(7.32)andobtain,bythedefinitionofRh,u
(t),v+a(u(t),v)=u
(t),v+a(Ru(t),v)=b(t,v).h0hh0hhhHereb(t,·)isasdefinedin(7.43).Subtractingthisequationfrom(7.44),weget

uh(t),vh0−u(t),vh0+a(θ(t),vh)=0,andthus=>

d
θ(t),vh0+a(θ(t),vh)=u(t),vh0−Rhu(t),vh=−(t),vh0.dt0TheapplicationofLemma7.4yields
tθ(t)≤θ(0)e−αt+
(s)e−α(t−s)ds.0000Sincetheellipticprojectioniscontinuous(Theorem7.9,(i))andu(t)issufficientlysmooth,Randthetimederivativedcommute;thatis,
(t)=hdt 7.2.SemidiscretizationbyVerticalMethodofLines305(R−I)u
(t).Itremainstoapplythetriangleinequalitytogetthestatedhresult.
Theorem7.10hasthefollowinginterpretation:Theerrornormuh(t)−u(t)0isestimatedby•theinitialerror(exponentiallydecayingint),whichoccursonlyifu0hdoesnotcoincidewiththeellipticprojectionofu0,•theprojectionerroroftheexactsolutionu(t)measuredinthenormofH,•theprojectionerrorofu
(t)measuredinthenormofHandintegrallyweightedbythefactore−α(t−s)on(0,t).Remark7.11Ifthebilinearformadefinesanellipticproblemsuchthatfortheellipticprojectionanerrorestimateofthetype22(I−Rh)w0≤Chw2forallw∈V∩H(Ω)isvalid,ifu0happroximatestheellipticprojectionRhu0oftheinitialvalueu0atleastwiththesameasymptoticquality,andifthesolutionuof(7.44)issufficientlysmooth,thenanoptimalL2-errorestimateresults:u(t)−u(t)≤C(u(t))h2.h0Weseethatinordertoobtainsemidiscreteerrorestimates,weneedesti-matesoftheprojectionerrormeasuredinthenormofH=L2(Ω).Dueto·0≤·V,thequasi-optimalityofRh(Theorem7.9,(ii))inconjunctionwiththecorrespondingapproximationerrorestimates(Theorem3.29)al-readyyieldsomeerrorestimate.Unfortunately,thisresultisnotoptimal.However,iftheadjointboundaryvalueproblemisregularinthesenseofDefinition3.36,thedualityargument(Theorem3.37)canbesuccessfullyusedtoderiveanoptimalresult.Theorem7.12SupposethebilinearformaisV-ellipticandcontinuous,andthesolutionoftheadjointboundaryvalueproblemisregular.Furthermore,letthespaceVh⊂Vbesuchthatforanyfunctionw∈V∩H2(Ω),infw−vhV≤Ch|w|2,vh∈VhwheretheconstantC>0doesnotdependonhandw.Ifu∈V∩H2(Ω),0thenforasufficientlysmoothsolutionuof(7.44)wehave−αtuh(t)−u(t)0≤u0h−u00e
t+Ch2ue−αt+u(t)+u
(s)e−α(t−s)ds.02220 3067.DiscretizationofParabolicProblemsProof:ThefirsttermintheerrorboundfromTheorem7.10isestimatedbymeansofthetriangleinequality:u0h−Rhu00≤u0h−u00+(I−Rh)u00.Thentheprojectionerrorestimate(Theorem3.37,(1))yieldsthegivenboundsoftheresultingsecondtermaswellasoftheremainingtwotermsintheerrorboundfromTheorem7.10.
OrderofConvergenceEstimatesfortheFiniteVolumeMethodForsimplicitywerestrictattentiontopurehomogeneousDirichletcondi-tions(Γ3=∂Ω).Theideaissimilartotheproofgiveninthefiniteelementcase.However,herewewillmeetsomeadditionaldifficulties,whicharecausedbytheuseofperturbedbilinearandlinearforms.Wetakev=vh∈Vhin(7.32)andsubtracttheresultfrom(7.48):u
(t),v−u
(t),v+a(u(t),v)−a(u(t),v)hh0,hh0hhhh=f(t),vh0,h−f(t),vh0.Inanalogytothefiniteelementmethod,weintroducethefollowingaux-iliaryproblem:Givensomev∈V,findanelementRhv∈Vhsuchthatah(Rhv,vh)=a(v,vh)forallvh∈Vh.(7.63)Withthis,theaboveidentitycanberewrittenasfollows:u
(t),v−u
(t),v+a(u(t)−Ru(t),v)hh0,hh0hhhh=f(t),vh0,h−f(t),vh0.34SubtractingfrombothsidesofthisrelationthetermdRu(t),vdthh0,handassumingthatu
(t)isasufficientlysmoothfunctionofx,aslightrearrangementyieldsθ
(t),v+a(θ(t),v)=−
(t),v+u
(t),v(7.64)h0,hhhh0,hh0
−u(t),vh0,h+f(t),vh0,h−f(t),vh0,where,asinthefiniteelementcase,θ(t)=uh(t)−Rhu(t)and(t)=Ru(t)−u(t).Furthermore,wedefine,forv∈V,b(t,v):=u
(t),v−hh10u
(t),vandb(t,v):=f(t),v−f(t),v.0,h20,h0Inordertobeabletoapplythediscretestabilityestimate(7.50)tothissituation,weneedanerrorestimateforRu
(t)asinRemark7.11andhbounds(consistencyerrorestimates)for|b1(t,v)|,|b2(t,v)|.Soweturntothefirstproblem.Infact,theestimateisverysimilartotheerrorestimateforthefinitevolumemethodgivenintheproofofTheorem6.18.Foranarbitraryfunctionv∈V∩H2(Ω)andv:=Rv−I(v),wehavehhhby(7.63)thatah(vh,vh)=ah(Rhv,vh)−ah(Ih(v),vh)=a(v,vh)−ah(Ih(v),vh). 7.2.SemidiscretizationbyVerticalMethodofLines307Bypartialintegrationinthefirsttermoftheright-handside,itfollowsthatah(vh,vh)=Lv,vh0−ah(Ih(v),vh).From[40]anestimateoftheright-handsideisknown(cf.also(6.22));thus221/2ah(vh,vh)≤Chv2|vh|1+vh0,h.SoTheorem6.15yields221/2|vh|1+vh0,h≤Chv2.Bythetriangleinequality,(Rh−I)v0,h≤Rhv−Ih(v)0,h+Ih(v)−v0,h.Sincethesecondtermvanishesbythedefinitionsof·0,handIh,wegetinparticular(Rh−I)v0,h≤Chv2.(7.65)Remark7.13Incontrasttothefiniteelementcase(Remark7.11),thisestimateisnotoptimal.Toestimate|b1(t,v)|and|b2(t,v)|,weprovethefollowingresult.Lemma7.14Assumew∈C1(Ω)andv∈V.Then,ifthefinitevolumehpartitionofΩisaDonalddiagram,|w,v0,h−w,v0|≤Ch|w|1,∞v0,h.Proof:Westartwithasimplerearrangementoftheorderofsummation:Mw,v0,h=wjvjmj=wjvj|Ωj,K|,j=1K∈Thj:∂KajwhereΩj,K=Ωj∩intK.First,wewillconsidertheinnersum.ForanytriangleK∈ThwithbarycentreaS,K,wecanwritewjvj|Ωj,K|=[wj−w(aS,K)]vj|Ωj,K|j:∂Kajj:∂Kaj0
1+w(aS,K)vj|Ωj,K|−vdxΩj,Kj:∂Kaj

+[w(aS,K)−w]vdx+wvdxΩj,KΩj,Kj:∂Kajj:∂Kaj
=:I1,K+I2,K+I3,K+wvdx.K 3087.DiscretizationofParabolicProblemsToestimateI1,K,weapplytheCauchy–Schwarzinequalityandget1/22|I1,K|≤|wj−w(aS,K)||Ωj,K|v0,h,K,j:∂Kajwhere1/22v0,h,K:=vj|Ωj,K|.j:∂KajSince|wj−w(aS,K)|≤hK|w|1,∞,itfollowsthat"|I1,K|≤hK|w|1,∞|K|v0,h,K.Similarly,forI3,Kweeasilyget
|I3,K|=[w(aS,K)−w]vdxΩK"≤w(aS,K)−w0,Kv0,K≤hK|w|1,∞|K|v0,K.SoitremainstoconsiderI2,K.Obviously,
I2,K=w(aS,K)[vj−v]dx.Ωj,Kj:∂KajWewillshowthatifΩjbelongstoaDonalddiagram,thenthesumvan-ishes.Todoso,letussupposethatthetriangleunderconsiderationhastheverticesai,aj,andak.ThesetΩj,Kcanbedecomposedintotwosub-trianglesbydrawingastraightlinebetweenaS,Kandaj.WewilldenotetheinteriorofthesetrianglesbyΩj,K,iandΩj,K,k;i.e.,Ωj,K,i:=intconv{aj,aS,K,aij},Ωj,K,k:=intconv{aj,aS,K,akj}.Oneachsubtriangle,theintegralofvcanbecalculatedexactlybymeansofthetrapezoidalrule.Since|Ωj,K,i|=|Ωj,K,k|=|K|/6inthecaseoftheDonalddiagram(cf.also(6.4)),wehave
AB|K|vj+vivj+vi+vkvdx=vj++Ωj,K,i1823AB|K|1151=vj+vi+vk,18663
AB|K|1151vdx=vj+vk+vi.Ω18663j,K,kConsequently,
AB|K|1177vdx=vj+vi+vk,Ωj,K18366 7.2.SemidiscretizationbyVerticalMethodofLines309andthus
|K|vdx=vj.Ωj,K3j:∂Kajj:∂KajOntheotherhand,since3|Ωj,K|=|K|(cf.(6.4)),wehave
|K|vjdx=vj,Ωj,K3j:∂Kajj:∂KajandsoI2,K=0.Insummary,wehaveobtainedthefollowingestimate:"&'|I1,K+I2,K+I3,K|≤hK|w|1,∞|K|v0,h,K+v0,K.Soitfollowsthat|w,v0,h−w,v0|≤|I1,K+I2,K+I3,K|K∈Th"&'≤h|w|1,∞|K|v0,h,K+v0,K.K∈ThBytheCauchy–Schwarzinequality,$%1/2$%1/2""2|K|v0,h,K≤|K|v0,h,K=|Ω|v0,hK∈ThK∈ThK∈Thand,similarly,""|K|v0,K≤|Ω|v0.K∈ThSowefinallyarriveat&'|w,v0,h−w,v0|≤Ch|w|1,∞v0,h+v0.Sincethenorms·0,hand·0areequivalentonVh(seeRemark6.16),weget|w,v0,h−w,v0|≤Ch|w|1,∞v0,h.
Nowwearepreparedtoapplythediscretestabilityestimate(7.50)toequation(7.64):θ(t)≤θ(0)e−αt0,h0,h
t+[
(s)+b(s)+b(s)]e−α(t−s)ds,0,h1∗2∗0where|bj(t,v)|≤bj(t)∗v0,hforallv∈Vh,t∈(0,T),andj=1,2.Thefirsttermintheintegralcanbeestimatedbymeansof(7.65),whereasthe 3107.DiscretizationofParabolicProblemsestimatesofb1(s)∗,b2(s)∗resultfromLemma7.14:θ(t)≤θ(0)e−αt0,h0,h
t+Ch[u
(s)+|u
(s)|+f(s)]e−α(t−s)ds.21,∞1,∞0Ifu∈V∩H2(Ω),wecanwrite,by(7.65),0θ(0)0,h≤uh0−u00,h+(I−Rh)u00,h≤uh0−u00,h+Chu02.SowegetAθ(t)≤u−ue−αt+Chue−αt0,hh000,h02
tB

−α(t−s)+[u(s)2+|u(s)|1,∞+f(s)1,∞]eds.0Sinceuh(t)−u(t)0,h≤θ(t)0,h+(Rh−I)u(t)0,h,theobtainedestimateand(7.65)yieldthefollowingresult.Theorem7.15InadditiontotheassumptionsofTheorem6.15,letf∈C([0,T],C1(Ω)),u∈V∩H2(Ω),andu∈V.Thenifu(t)issufficiently00hhsmooth,thesolutionuh(t)ofthesemidiscretefinitevolumemethod(7.48)onDonalddiagramssatisfiesthefollowingestimate:A−αt−αtuh(t)−u(t)0,h≤uh0−u00,he+Chu02e+u(t)2
tB+[u
(s)+|u
(s)|+f(s)]e−α(t−s)ds.21,∞1,∞0Remark7.16Incomparisonwiththefiniteelementmethod,theresultisnotoptimalinh.Thereasonisthat,ingeneral,thefinitevolumemethoddoesnotyieldoptimalL2-errorestimatesevenintheellipticcase,butthistypeofresultisnecessarytoobtainoptimalestimates.Exercises7.4LetA∈RM,MbeanRM-ellipticmatrixandletthesymmetricpositivedefinitematrixB∈RM,MhavetheCholeskydecompositionB=ETE.ShowthatthematrixAˆ:=E−TAE−1isRM-elliptic.7.5Proveidentity(7.62)byfirstprovingthecorrespondingidentityforonespacedimension:l−1πahsin2νih=.a2i=1 7.3.FullyDiscreteSchemes3117.6LetVbeaBanachspaceanda:V×V→RaV-elliptic,continuousbilinearform.ShowthattheRitzprojectionRh:V→VhinasubspaceVh⊂V(cf.Definition7.8)hasthefollowingproperties:(i)R:V→ViscontinuousbecauseofRu≤Mu,hhhVαV(ii)Rhyieldsquasi-optimalapproximations;thatis,Mu−RhuV≤infu−vhV.αvh∈VhHereMandαdenotetheconstantsinthecontinuityandellipticityconditions,respectively.7.7Letu∈C1([0,T],V).ShowthatRu∈C1([0,T],V)anddRu(t)=hdthRdu(t).hdt7.8TransferthederivationofthefinitevolumemethodgiveninSec-tion6.2.2forthecaseofanellipticboundaryvalueproblemtotheparabolicinitial-boundaryvalueproblem(7.1)indivergenceform;i.e.,convinceyour-selfthattheformalismofobtaining(7.48)indeedcanbeinterpretedasafinitevolumesemidiscretizationof(7.1).7.3FullyDiscreteSchemesAswehaveseen,theapplicationoftheverticalmethodoflinesresultsinthefollowingsituation:•Thereisalinearsystemofordinarydifferentialequationsofhighorder(dimension)tobesolved.•Thereisanerrorestimateforthesolutionuoftheinitial-boundaryvalueproblem(7.1)bymeansoftheexactsolutionuhofthesystem(7.41).Adifficultyinthechoiceandintheanalysisofanappropriatediscretiza-tionmethodforsystemsofordinarydifferentialequationsisthatmanystandardestimatesinvolvetheLipschitzconstantofthecorrespondingright-handside,hereqh−Ahuh(cf.(7.41)).Butthisconstantistypicallylargeforsmallspatialparametersh,andsowewouldobtainnonrealisticerrorestimates(cf.Theorem3.45).Therearetwoalternatives.Forcomparativelysimpletimediscretiza-tions,certainestimatescanbederivedinadirectway(i.e.,withoutusingstandardestimatesforsystemsofordinarydifferentialequations).Thesec-ondwayistoapplyspecifictimediscretizationsinconjunctionwithrefinedmethodsofproof. 3127.DiscretizationofParabolicProblemsHerewewillexplainthefirstwayfortheso-calledone-step-thetamethod.One-StepDiscretizationsinTime,inParticularfortheFiniteDifferenceMethodWestartfromtheproblem(7.41),whichresultedfromspatialdiscretizationtechniques.ProvidedthatT<∞,thetimeinterval(0,T)issubdividedintoN∈Nsubintervalsofequallengthτ:=T/N.Furthermore,wesett:=nτforn∈{0,...,N}andun∈RMforanapproximationofu(t).nhhnIfthetimeintervalisunbounded,thetimestepτ>0isgiven,andthenumbern∈Nisallowedtoincreasewithoutbounded;thatis,wesetformallyN=∞.Thevaluest=tn,whereanapproximationistobedetermined,arecalledtimelevels.Therestrictiontoequidistanttimestepsisonlyforthesakeofsimplicity.Weapproximatedubythedifferencequotientdthd1uh(t)∼(uh(tn+1)−uh(tn)).dtτIfweinterpretthisapproximationtobeatt=tn,wetaketheforwarddifferencequotient;att=tn+1wetakethebackwarddifferencequotient;att=t+1τwetakethesymmetricdifferencequotient.Againweobtainn2ageneralizationandunificationbyintroducingaparameterΘ∈[0,1]andinterpretingtheapproximationtobetakenatt=tn+Θτ.AsforΘ=0or1,wearenotatatimelevel,andsoweneedthefurtherapproximationAhuh((n+Θ)τ)∼ΘAhuh(tn)+ΘAhuh(tn+1).HereweusetheabbreviationΘ:=1−Θ.The(one-step-)thetamethoddefinesasequenceofvectorsu0,...,uNby,forn=0,1,...,N−1,hh1un+1−un+ΘAun+ΘAun+1=q((n+Θ)τ),(7.66)τhhhhhhhu0=u.h0Ifweapplythisdiscretizationtothemoregeneralform(7.45),wegetcorrespondingly1Bun+1−Bun+ΘAˆun+ΘAˆun+1=q((n+Θ)τ).(7.67)τhhhhhhhhhAnalogouslyto(7.45),themoregeneralformcanbetransformedto(7.66),−1assumingthatBhisregular:eitherbymultiplying(7.67)byBhorinthecaseofadecompositionB=ETE(forasymmetricpositivedefiniteB)hhhhbymultiplyingbyE−TandachangeofvariablestoEun.Wewillapplyhhhtwotechniquesinthefollowing:OneisbasedontheeigenvectordecompositionofAh;thusfor(7.67),thismeanstoconsiderthegeneralizedeigenvalueproblemAˆhv=λBhv.(7.68) 7.3.FullyDiscreteSchemes313NotethattheGalerkinapproachfortheeigenvalueproblemsaccordingtoDefinition7.6leadstosuchageneralizedeigenvalueproblemwiththestiffnessmatrixAˆhandthemassmatrixBh.Theotherapproachisbasedonthematrixproperties(1.32)*or(1.32).Forthemostimportantcase,Bh=diag(bi),bi>0fori=1,...,M,(7.69)whichcorrespondstothemasslumpingprocedure,theabove-mentionedtransformationreducestoadiagonalscaling,whichdoesnotinfluenceanyoftheirproperties.Havingthisinmind,inthefollowingwewillconsiderexplicitlyonlytheformulation(7.66).InthecaseΘ=0,theexplicitEulermethod,uncanbedeterminedhexplicitlybyun+1=τ(q(t)−Aun)+un=(I−τA)un+τq(t).hhnhhhhhnThustheeffortforonetimestepconsistsofaSAXPYoperation,avectoraddition,andamatrix-vectoroperation.FordimensionMthefirstoftheseisofcomplexityO(M),andalsothelastoneifthematrixissparseinthesensedefinedatthebeginningofChapter5.Ontheotherhand,forΘ=0,themethodisimplicit,sinceforeachtimestepasystemoflinearequationshastobesolvedwiththesystemmatrixI+ΘτAh.HerethecasesΘ=1,theimplicitEulermethod,andΘ=1,theCrank–Nicolsonmethod,will2beofinterest.Duetoourrestrictiontotime-independentcoefficients,thematrixisthesameforeverytimestep(forconstantτ).Ifdirectmethods(seeSection2.5)areused,thentheLUfactorizationhastobecomputedonlyonce,andonlyforwardandbackwardsubstitutionswithchangingright-handsidesarenecessary,wherecomputationforΘ=1alsorequiresamatrix-vectoroperation.Forbandmatrices,forexample,operationsofthecomplexitybandwidth×dimensionarenecessary,whichmeansforthebasicexampleoftheheatequationonarectangleO(M3/2)operationsin-steadofO(M)fortheexplicitmethod.Iterativemethodsfortheresolutionof(7.66)cannotmakeuseoftheconstantmatrix,butwithunthereisahgoodinitialiterateifτisnottoolarge.AlthoughtheexplicitEulermethodΘ=0seemstobeattractive,wewillseelaterthatwithrespecttoaccuracyorstabilityonemaypreferΘ=12orΘ=1.Toinvestigatefurtherthethetamethod,weresolverecursivelytherelations(7.66)togaintherepresentationnn−1uh=(I+ΘτAh)I−ΘτAhu0(7.70)nn−k−1−1+τ(I+ΘτAh)I−ΘτAh(I+ΘτAh)qhtk−Θτ.k=1 3147.DiscretizationofParabolicProblemsHereweusetheabbreviationA−n=(A−1)nforamatrixA.Compar-ingthiswiththesolution(7.55)ofthesemidiscreteproblem,weseetheapproximationse−Ahtn∼En,h,τwhere−1Eh,τ:=(I+ΘτAh)I−ΘτAhand
tn
tn−Ah(tn−s)−Ahτ(tn−s)/τeqh(s)ds=eqh(s)ds00n(tn−s)/τ−1∼τEh,τ(I+ΘτAh)qhs−Θτ.k=1s=kτ(7.71)ThematrixEh,τthusisthesolutionoperatorof(7.66)foronetimestepandhomogeneousboundaryconditionsandright-handside.Itistobeexpectedthatithastocapturethequalitativebehaviourofe−Ahτthatitisapproximating.Thiswillbeinvestigatedinthenextsection.One-StepDiscretizationsfortheFiniteElementMethodThefullydiscreteschemecanbeachievedintwoways:Besidesapply-ing(7.66)to(7.41)inthetransformedvariableorintheform(7.67),thediscretizationapproachcanapplieddirectlyto(7.44):With∂Un+1:=(Un+1−Un)/τ,fn+s:=sf(t)+(1−s)f(t),n+1nbn+s(v):=sb(t,v)+(1−s)b(t,v),baccordingto(7.43),s∈[0,1],n+1nandwithafixednumberΘ∈[0,1],thefullydiscretemethodfor(7.44)thenreadsasfollows:FindasequenceU0,...,UN∈Vsuchthatforn∈{0,...,N−1},h34∂Un+1,v+a(ΘUn+1+ΘUn,v)=bn+Θ(v)h0hhforallvh∈Vh,(7.72)U0=u.0hAnalternativechoicefortheright-handside,closertothefinitedifferencemethod,isthedirectevaluationattn+Θτ,e.g.,f(tn+Θτ).TheversionhereischosentosimplifytheorderofconvergenceestimateinSection7.6.ByrepresentingtheUnbymeansofabasisofVasafter(7.44),againhwegettheform(7.67)(or(7.66)inthetransformedvariable).NotethatalsoforΘ=0theproblemhereisimplicitifBhisnotdiagonal.Therefore,masslumpingisoftenapplied,andthescalarproduct·,·0in(7.72)is 7.4.Stability315replacedbyanapproximationduetonumericalquadrature,i.e.,∂Un+1,v+a(ΘUn+1+ΘUn,v)=bn+Θ(v)h0,hhhforallvh∈Vh,(7.73)U0=u.0hAsexplainedinSection3.5.2,uh,vh0,histhesumoverallcontributionsfromelementsK∈Th,whichtakestheform(3.112)forthereferenceelement.InthecaseofLagrangeelementsandanodalquadratureformulawehaveforthenodalbasisfunctionsϕi:ϕj,ϕi0,h=ϕi,ϕi0,hδij=:biδijfori,j=1,...,M,(7.74)sincefori=jtheintegrandϕiϕjvanishesatallquadraturepoints.Inthiscasewearriveattheform(7.67)withamatrixBhsatisfying(7.69).One-StepDiscretizationsfortheFiniteVolumeMethodAsintheprevioussubsectiononthefiniteelementapproach,thesemidiscreteformulation(7.48)canbediscretizedintimedirectly:FindasequenceU0,...,UN∈Vsuchthatforn∈{0,...,N−1},h3434∂Un+1,v+a(ΘUn+1+ΘUn,v)=fn+Θ,vh0,hhhh0,hforallvh∈Vh,(7.75)U0=u,0hwhere∂Un+1,Θ,fn+Θaredefinedasbefore(7.72).Rememberthathereweconsideronlyhomogeneousboundaryconditions.IftheelementsUn,Un+1arerepresentedbymeansofabasisofV,wehrecovertheform(7.67).SincethemassmatrixBhisdiagonal,theproblemcanberegardedasbeingexplicitforΘ=0.Exercise7.9ConsiderlinearsimplicialelementsdefinedonageneralconformingtriangulationofapolygonallyboundeddomainΩ⊂R2.(a)DeterminetheentriesofthemassmatrixBh.(b)Usingthetrapezoidalrule,determinetheentriesofthelumpedmassmatrixdiag(bi).7.4StabilityInSection7.3wehaveseenthatatleastifabasisofeigenvectorsofthediscretizationmatrixAhallowsforthesolutionrepresentation(7.55)for 3167.DiscretizationofParabolicProblemsthesemidiscretemethod,thequalitativebehaviourofe−Ahτushouldbe0preservedbyEh,τu0,beingonetimestepτforhomogeneousboundaryconditionsandright-handside(qh=0)inthesemi-andfullydiscretecases.Itissufficienttoconsidertheeigenvectorwiinsteadofageneralu0.Thus,wehavetocomparee−Ahτw=(e−λiτ)w(7.76)iiwith−11−Θτλi(I+ΘτAh)I−ΘτAhwi=wi.(7.77)1+ΘτλiWeseethattheexponentialfunctionisapproximatedby1+(1−Θ)zR(z)=,(7.78)1−Θzthestabilityfunction,atthepointsz=−λiτ∈C,givenbytheeigenvaluesλi,andthetimestepsizeτ.Forntimestepsandqh=0wehavene−Ahτw=e−λitnw∼R(−λτ)nw.(7.79)iiiiThus,therestrictiontoeigenvectorswiwitheigenvaluesλihasdiagonal-izedthesystemofordinarydifferentialequations(7.41)forqh=0tothescalarproblems
ξ+λiξ=0,t∈(0,T),(7.80)ξ(0)=ξ0(forξ=1)withitssolutionξ(t)=e−λitξ0,forwhichtheone-step-theta0methodgivestheapproximationξ=R(−λτ)ξ=(R(−λτ))n+1ξ(7.81)n+1ini0att=tn+1.Abasicrequirementforadiscretionmethodisthefollowing:Definition7.17Aone-stepmethodiscallednonexpansiveiffortwonu-mericalapproximationsunandu˜n,generatedunderthesameconditionshhexceptfortwodiscreteinitialvaluesu0andu˜0,respectively,thefollowingestimateisvalid:|un+1−u˜n+1|≤|un−u˜n|,n∈{0,...,N−1}.hhArecursiveapplicationofthisestimateimmediatelyresultsin|un−u˜n|≤|u−u˜|,n∈{1,...,N}.00Hereageneralone-stepmethodhastheformun+1=un+τΦ(τ,t,un),n∈{0,...,N−1},hhnhwithu0=uandaso-calledgeneratingfunctionΦ:R×[0,T)×RM→h0+RMthatcharacterizestheparticularmethod.Thegeneratingfunctionof 7.4.Stability317theone-step-thetamethodappliedtothesystem(7.41)is−1Φ(τ,t,ξ)=−(I+τΘAh)[Ahξ−qh(t+Θτ)].Thusnonexpansivenessmodelsthefactthatperturbances,i.e.,inpartic-ularerrors,arenotamplifiedintimebythenumericalmethod.Thisisconsiderablyweakerthantheexponentialdecayinthecontinuoussolution(see(7.18)),whichwouldbetoostrongarequest.Havinginmind(7.79)–(7.81),andexpectingthe(realpartsofthe)ei-genvaluestobepositive,thefollowingrestrictionissufficient:Definition7.18Aone-stepmethodiscalledA-stableifitsapplicationtothescalarmodelproblem(7.80)ξ
+λξ=0,t∈(0,T),ξ(0)=ξ0,yieldsanonexpansivemethodforallcomplexparametersλwithλ>0andarbitrarystepsizesτ>0.Becauseof(7.81)wehaveξn+1−ξ˜n+1=R(−λτ)[ξn−ξ˜n]fortwoapproximationsoftheone-step-thetamethodappliedto(7.80).Thisshowsthatthecondition|R(z)|≤1forallzwithz<0issufficientfortheA-stabilityofthemethod.Moregenerally,anyone-stepmethodthatcanbewrittenfor(7.80)intheformξn+1=R(−λiτ)ξn(7.82)isnonexpansiveiff|R(−λiτ)|≤1.(7.83)Theone-step-thetamethodisnonexpansivefor(7.41)inthecaseofaneigenvectorbasisif(7.83)holdsforalleigenvaluesλiandstepsizeτ.Aconvenientformulationcanbeachievedbythenotionofthedomainofstability.Definition7.19GivenastabilityfunctionR:C→C,thesetSR:={z∈C:|R(z)|<1}iscalledadomainof(absolute)stabilityoftheone-stepmethodξn+1=R(−λτ)ξn.Example7.20Fortheone-step-thetamethodwehave:(1)ForΘ=0,SRisthe(open)unitdiskwithcentrez=−1. 3187.DiscretizationofParabolicProblems(2)ForΘ=1,Scoincideswiththeleftcomplexhalf-plane(exceptfor2Rtheimaginaryaxis).(3)ForΘ=1,SRisthewholecomplexplaneexceptfortheclosedunitdiskwithcentrez=1.ThenotionofA-stabilityreflectsthefactthattheproperty|e−λτ|≤1forλ>0issatisfiedbythefunctionR(−λτ),too:Corollary7.21ForacontinuousstabilityfunctionRtheone-stepmethodξn+1=R(−λτ)ξnisA-stableiftheclosureSofitsdomainofstabilityRcontainstheleftcomplexhalf-plane.ThustheCrank–NicolsonandtheimplicitEulermethodsareA-stable,butnottheexplicitEulermethod.Tohavenonexpansiveness,weneedtherequirement|1−λiτ|=|R(−λiτ)|≤1,(7.84)whichisastepsizerestriction:Forpositiveλiitreadsτ≤2/max{λi|i=1,...,M}.(7.85)Fortheexampleofthefive-pointstencildiscretizationoftheheatequationonarectanglewithDirichletboundaryconditionsaccordingto(7.37)–(7.39),equation(7.84)readsτππ1−22−cosνh−cosµh≤1(7.86)h2abforallν=1,...,l−1,µ=1,...,m−1.Thefollowingconditionissufficient(andforl,m→∞alsonecessary):τ1≤.(7.87)h24Forthefiniteelementmethodasimilarestimateholdsinamoregeneralcontext.UndertheassumptionsofTheorem3.45weconcludefromitsproof(see(3.141))thatthefollowingholds:−2max{λi|i=1,...,M}≤CminhKK∈ThfortheeigenvaluesofB−1Aˆ,whereB=ETEisthemassmatrixandhhhhhAˆthestiffnessmatrix,andthusalsoforA=EB−1AˆE−1.HereCishhhhhhaconstantindependentofh.Therefore,wehave2τminhK≤2/C(7.88)K∈ThasasufficientconditionforthenonexpansivenessofthemethodwithaspecificconstantdependingonthestabilityconstantofthebilinearformandtheconstantfromTheorem3.43,(2). 7.4.Stability319ThesestepsizerestrictionsimpedetheattractivityoftheexplicitEulermethod,andsoimplicitversionsareoftenused.ButalsointheA-stablecasetherearedistinctionsinthebehaviour(ofthestabilityfunctions).Comparingthem,weseethatforΘ=1:R(−x)→−1forx→∞;2(7.89)forΘ=1:R(−x)→0forx→∞.ThismeansthatfortheimplicitEulermethodtheinfluenceoflargeeigen-valueswillbemoregreatlydamped,thelargertheyare,correspondingtotheexponentialfunctiontobeapproximated,buttheCrank–Nicolsonmethodpreservesthesecomponentsnearlyundampedinanoscillatorymanner.Thismayleadtoaproblemfor“rough”initialdataordiscontinu-itiesbetweeninitialdataandDirichletboundaryconditions.Ontheotherhand,theimplicitEulermethodalsomaydampsolutioncomponentstoostrongly,makingthesolution“too”smooth.Thecorrespondingnotionisthefollowing:Definition7.22One-stepmethodswhosestabilityfunctionsatisfiesR(z)→0forz→−∞,arecalledL-stable.AnintermediatepositionisfilledbythestronglyA-stablemethods.Theyarecharacterizedbytheproperties•|R(z)|<1forallzwithz<0,•lim|R(z)|<1.z→−∞Example7.23(1)Amongtheone-step-thetamethods,onlytheimplicitEulermethod(Θ=1)isL-stable.(2)TheCrank–Nicolsonmethod(Θ=1)isnotstronglyA-stable,2becauseof(7.89).Thenonexpansivenessofaone-stepmethodcanalsobecharacterizedbyanormconditionforthesolutionoperatorEh,τ.Theorem7.24LetthespatialdiscretizationmatrixAhhaveabasisofeigenvectorswiorthogonalwithrespecttothescalarproduct·,·h,witheigenvaluesλi,i=1,...,M.Considertheproblem(7.41)anditsdiscretizationintimebyaone-stepmethodwithalinearsolutionrepresentationun=Enu(7.90)hh,τ0forq=0,whereE∈RM,M,andastabilityfunctionRsuchthat(7.82)hh,τandEh,τwi=R(−λiτ)wi(7.91) 3207.DiscretizationofParabolicProblemsfori=1,...,M.Thenthefollowingstatementsareequivalent:(1)Theone-stepmethodisnonexpansiveforthemodelproblem(7.80)andalleigenvaluesλiofAh.(2)Theone-stepmethodisnonexpansivefortheproblem(7.41),withrespecttothenorm·hinducedby·,·h.(3)Eh,τh≤1inthematrixnorm·hinducedbythevectornorm·h.Proof:Weprove(1)⇒(3)⇒(2)⇒(1):(1)⇒(3):Accordingto(7.83)(1)ischaracterizedby|R(−λiτ)|≤1,(7.92)fortheeigenvaluesλi.Fortheeigenvector,wiwitheigenvalueλiwehave(7.91),andthus,forManarbitraryu0=i=1ciwi,MEu2=cEw2h,τ0hih,τihi=1MM=cR(−λτ)w2=c2|R(−λτ)|2w2,iiihiiihi=1i=1becauseoftheorthogonalityofthewi,andanalogously,Mu2=c2w2,0hiihi=1andfinally,becauseof(7.92),MEu2≤c2w2=u2,h,τ0hiih0hi=1whichisassertion(3).(3)⇒(2):isobvious.(2)⇒(3):|R(−λiτ)|wih=R(−λiτ)wih=Eh,τwih≤wih.
Thus,nonexpansivenessisoftenidenticaltowhatis(vaguely)calledstability:Definition7.25Aone-stepmethodwithasolutionrepresentationEh,τforqh=0iscalledstablewithrespecttothevectornorm·hifEh,τh≤1intheinducedmatrixnorm·h. 7.4.Stability321Tillnowwehaveconsideredonlyhomogeneousboundarydataandright-handsides.Atleastfortheone-step-thetamethodthisisnotarestriction:Theorem7.26Considertheone-step-thetamethodundertheassumptionofTheorem7.24,withλi≥0,i=1,...,M,andwithτsuchthatthemethodisstable.Thenthesolutionisstableininitialconditionu0andright-handsideqhinthefollowingsense:nun≤u+τqt−Θτ.(7.93)hh0hhkhk=1Proof:Fromthesolutionrepresentation(7.70)weconcludethatnun≤Enu+τEn−k(I+τΘA)−1q(t−Θτ)hhh,τh0hh,τhhhhkhk=1(7.94)usingthesubmultiplicativityofthematrixnorm.Wehavetheestimate1(I+ΘτA)−1w=w≤w,hih1+ΘτλihihiandthusasintheproofofTheorem7.24,(1)⇒(3),(I+ΘτA)−1≤1hhconcludestheproof.
ThestabilityconditionrequiresstepsizerestrictionsforΘ<1,which2havebeendiscussedaboveforΘ=0.TherequirementofstabilitycanbeweakenedtoEh,τh≤1+Kτ(7.95)forsomeconstantK>0,whichinthesituationofTheorem7.24isequi-valentto|R(−λτ)|≤1+Kτ,foralleigenvaluesλofAh.Becauseof(1+Kτ)n≤exp(Knτ),in(7.93)theadditionalfactorexp(KT)appearsandcorrespondinglyexp(K(n−k)τ)inthesum.Iftheprocessistobeconsideredonlyinasmalltimeinterval,thisbecomespartoftheconstant,butforlargetimehorizonstheestimatebecomesinconclusive.Ontheotherhand,fortheone-step-thetamethodfor1<Θ≤1the2estimateEh,τh≤1andthustheconstantsin(7.93)canbesharpenedtoEh,τh≤R(−λminτ),whereλministhesmallesteigenvalueofAh, 3227.DiscretizationofParabolicProblemsreflectingtheexponentialdecay.Forexample,forΘ=1,the(errorinthe)initialdataisdampedwiththefactornn1Eh,τh=R(−λminτ)=n,(1+λminτ)whichforτ≤τ0forsomefixedτ0>0canbeestimatedbyexp(−λnτ)forsomeλ>0.Weconcludethissectionwithanexample.Example7.27(Prothero-Robinsonmodel)Letg∈C1[0,T]begiven.Weconsidertheinitialvalueproblemξ
+λ(ξ−g)=g
,t∈(0,T),ξ(0)=ξ0.Obviously,gisaparticularsolutionofthedifferentialequation,sothegeneralsolutionisξ(t)=e−λt[ξ−g(0)]+g(t).0Inthespecialcaseg(t)=arctant,λ=500,andfortheindicatedvaluesofξ0,Figure7.1showsthequalitativebehaviourofthesolution.400500-100Figure7.1.Prothero–Robinsonmodel.Itisworthmentioningthatthefigureisextremelyscaled:Thecontinuousline(toξ0=0)seemstobestraight,butitisthegraphofg.TheexplicitEulermethodforthismodelisn+1n
ξ=(1−λτ)ξ+τ[g(tn)+λg(tn)]. 7.5.TheMaximumPrinciplefortheOne-Step-ThetaMethod323Accordingtotheaboveconsiderations,itisnonexpansiveonlyifλτ≤1holds.Forlargenumbersλ,thisisaveryrestrictivestepsizecondition;seealsothediscussionof(7.85)to(7.87).Duetotheirbetterstabilityproperties,implicitmethodssuchastheCrank–NicolsonandtheimplicitEulermethodsdonothavesuchstepsizerestrictions.Nevertheless,theapplicationofimplicitmethodsisnotfreefromsurprises.Forexample,inthecaseoflargenumbersλ,anorderreductioncanoccur.Exercises7.10DeterminethecorrespondingdomainofstabilitySRoftheone-step-thetamethodforthefollowingvaluesoftheparameterΘ:0,1,1.27.11ShowtheL-stabilityoftheimplicitEulermethod.7.12(a)Showthatthediscretizationξn=ξn−2+2τf(t,ξn−1),n=2,...Nn−1(midpointrule),appliedtothemodelequationξ
=f(t,ξ)withf(t,ξ)=−λξandλ>0leads,forasufficientlysmallstepsizeτ>0,toageneralsolutionthatcanbeadditivelydecomposedintoadecayingandanincreasing(byabsolutevalue)oscillatingcomponent.(b)ShowthattheoscillatingcomponentcanbedampedifadditionallythequantityξNiscomputed(modifiedmidpointrule):∗1&'ξN=ξN+ξN−1+τf(t,ξN).∗N27.13Letm∈Nbegiven.FindapolynomialRm(z)=1+z+mjj=2γjz(γj∈R)suchthatthecorrespondingdomainofabsolutesta-bilityforR(z):=Rm(z)containsanintervalofthenegativerealaxisthatisaslargeaspossible.7.5TheMaximumPrinciplefortheOne-Step-ThetaMethodInSection1.4wehaveseenthatforadiscreteproblemoftheform(1.31)thereisahierarchyofpropertiesrangingfromacomparisonprincipletoastrongmaximumprinciple,whichisinturnappliedbyahierarchyofconditions,partlysummarizedas(1.32)or(1.32)∗.Toremindthereader,weregrouptheseconditionsaccordingly: 3247.DiscretizationofParabolicProblemsThecollectionofconditions(1.32),(1),(2),(3)i),(4)∗iscalled(IM).(IM)impliestheinversemonotonicityofAh(Theorem1.12,(1.39)).Thecollectionofconditions(IM),(5)iscalled(CP).(CP)impliesacomparisonprincipleinthesenseofCorollary1.13.Thecollectionofconditions(CP),(6)∗iscalled(MP)∗.(MP)∗impliesamaximumprincipleintheformofTheorem1.10(1.38).Alternatively,thecollectionofconditions(CP)(6)#(seeExercise1.13)iscalled(MP).(MP)impliesamaximumprincipleintheformofTheorem1.9(1.34).Finally,thecollectionofconditions(CP),(6),(4)(insteadof(4)∗),(7)iscalled(SMP).(SMP)impliesastrongmaximumprincipleinthesenseofTheorem1.9.AnL∞-stabilityestimateinthesenseofTheorem1.14iscloselyrelated.Thiswillbetakenupinthenextsection.Inthefollowingwewilldiscusstheabove-mentionedpropertiesfortheone-step-thetamethod,castintotheform(1.31),onthebasisofcorrespond-ingpropertiesoftheunderlyingellipticproblemanditsdiscretization.Itwillturnoutthatunderareasonablecondition(see(7.100)),condition(4)∗(andthus(3)ii))willnotbenecessaryfortheellipticproblem.Thisreflectsthefactthatcontrarytotheellipticproblem,fortheparabolicproblemalsothecaseofapureNeumannboundarycondition(wherenodegreesoffree-domaregivenandthuseliminated)isallowed,sincetheinitialconditionactsasaDirichletboundarycondition.Inassumingthatthediscretizationoftheunderlyingellipticproblemisoftheform(1.31),wereturntothenotationM=M1+M2,whereM2isthenumberofdegreesoffreedomeliminated,andthusA,B∈RM1,M1.hhWewritethediscreteproblemaccordingto(7.66)asonelargesystemofequationsfortheunknownu1hu2huh=.,(7.96)..uNhinwhichthevectorofgridvaluesui∈RM1arecollectedtoonelargehvectorofdimensionM1:=N·M1.ThusthegridpointsinΩ×(0,T)arethepoints(xj,tn),n=1,...,N,xj∈Ωh,e.g.,forthefinitedifferencemethod.ThedefiningsystemofequationshastheformChuh=ph,(7.97) 7.5.MaximumPrinciplefortheOne-Step-ThetaMethod325whereI+τΘAh..−I+τΘAh.0....Ch=..,....0..−I+τΘAhI+τΘAhagainwithΘ:=1−Θ,τqh(Θτ)+(I−τΘAh)u0τqh((1+Θ)τ).ph=......τqh(N−1+Θ)τ)Sincethespatialdiscretizationisperformedasinthestationarycase,andinthenthstepthediscretizationrelatestot=tn−1+ΘτandalsotheapproximationAhu(tn−1+Θτ)∼ΘAhu(tn−1)+ΘAhu(tn)enterstheformulation(7.66),wecanassumetohavethefollowingstructureoftheright-handsideof(7.66):n−1nqh((n−1+Θ)τ)=−Aˆh(Θuˆh+Θuˆh)+f((n−1+Θ)τ)forn=1,...,N.(7.98)Heretheuˆn∈RM2aretheknownspatialboundaryvaluesontimelevelhtn,whichhavebeeneliminatedfromtheequationasexplained,e.g.,inChapter1forthefinitedifferencemethod.Butasnoted,weallowalsoforthecasewheresuchvaluesdonotappear(i.e.,M2=0)then(7.98)reducestoqh((n−1+Θ)τ)=f((n−1+Θ)τ)forn=1,...,N.Forthecontinuousproblem,dataareprescribedattheparabolicboundaryiΩ×{0}∪∂Ω×[0,T];correspondingly,theknownvaluesuˆharecollectedwiththeinitialdatau∈RM1toalargevector0u0uˆ0huˆ1uˆh=h,...Nuˆh 3267.DiscretizationofParabolicProblemsi.e.,avectorofdimensionM2:=M1+(N+1)M2,whichmayreducetouˆh=u0∈RM1.Withthisnotationwehaveph=−Cˆhuˆh+e(7.99)ifwedefine−I+τΘAhτΘAˆhτΘAˆhO......CˆO...h=........,....OτΘAˆhτΘAˆhτf(Θτ)τf((1+Θ)τ).e=......τf((N−1+Θ)τ)Inthefollowingthevalidityof(1.32)∗or(1.32)forC˜h=(Ch,Cˆh)willbeinvestigatedonthebasisofcorrespondingpropertiesofA˜h=(Ah,Aˆh).NotethatevenifAhisirreducible,thematrixChisalwaysreducible,sinceundependsonlyonu1,...,un−1,butnotonthefuturetimelevels.hhh(Therefore,(7.97)servesonlyforthetheoreticalanalysis,butnotfortheactualcomputation.)InthefollowingweassumethatτΘ(Ah)jj<1forj=1,...,M1,(7.100)whichisalwayssatisfiedfortheimplicitEulermethod(Θ=1).Then:(1)(Ch)rr>0forr=1,...,M1holdsif(1)isvalidforAh.Actually,also(Ah)jj>−1/(τΘ)wouldbesufficient.(2)(Ch)rs≤0forr,s=1,...,M1,r=s:If(2)isvalidforAh,thenonlythenonpositivityofthediagonalelementsoftheoff-diagonalblockofCh,−I+τΘAh,isinquestion.Thisisensuredby(7.100)(weakenedto“≤”).M1(3)(i)Cr:=Ch≥0forr=1,...,M1:s=1rs 7.5.MaximumPrinciplefortheOne-Step-ThetaMethod327(ii)Cr>0foratleastoner∈{1,...,M1}:WesetM1Aj:=(Ah)jk,k=1sothatcondition(3)(i)forAhmeansthatAj≥0forj=1,...,M1.Therefore,wehaveCr=1+τΘAj>0(7.101)fortheindicesrofthefirsttimelevel,wherethe“global”indexrcorrespondstothe“local”spatialindexj.Forthefollowingtimelevels,therelationCr=1−1+τ(Θ+Θ)Aj=τAj≥0(7.102)holds,i.e.,(3)(i)and(ii).(4)∗Foreveryr∈{1,...,M}satisfying11M1(Ch)rs=0(7.103)r=1thereexistindicesr2,...,rl+1suchthat(Ch)riri+1=0fori=1,...,landM1(Ch)rl+1s>0.(7.104)s=1Toavoidtoomanytechnicalities,weadoptthebackgroundofafinitedifferencemethod.Actually,onlymatrixpropertiesenterthereason-ing.Wecall(space-time)gridpointssatisfying(7.103)farfromtheboundary,andthosesatisfying(7.104)closetotheboundary.Dueto(7.101),allpointsofthefirsttimelevelareclosetotheboundary(consistentwiththefactthatthegridpointsfort0=0belongtotheparabolicboundary).Forthesubsequenttimeleveln,dueto(7.102),apoint(xi,tn)isclosetotheboundaryifxiisclosetothebound-arywithrespecttoA˜.Therefore,therequirementof(4)∗,thatahpointfarfromtheboundarycanbeconnectedviaachainofneigh-bourstoapointclosetotheboundary,canberealizedintwoways:Firstly,withinthetimeleveln,i.e.,thediagonalblockofChifAhsatisfiescondition(4)∗.Secondly,withoutthisassumptionachainofneighboursexistby(x,tn),(x,tn−1)upto(x,t1),i.e.,apointclosetotheboundary,sincethediagonalelementof−I+τΘAhdoesnotvanishdueto(7.100).Thisreasoningadditionallyhasestablishedthefollowing: 3287.DiscretizationofParabolicProblems(4)#IfAisirreducible,thenagridpoint(x,t),x∈Ωcanbeconnectedhnhviaachainofneighbourstoeverygridpoint(y,tk),y∈Ωhand0≤k≤n.(5)(Cˆh)rs≤0forr=1,...,M1,s=M1+1,...,M2:Analogouslyto(2),thisfollowsfrom(5)forAˆhand(7.100).M(6)∗C˜:=(C˜)=0forr=1,...,M:rhrss=1Analogouslyto(7.102),wehaveMC˜r=τA˜j:=τ(A˜h)jk,k=1sothatthepropertyisequivalenttothecorrespondingoneofA˜h.(6)C˜r≥0forr=1,...,Misequivalentto(6)forA˜hbytheaboveargument.(7)Foreverys∈M1+1,...,Mthereexistsanr∈{1,...,M1}suchthat(Cˆh)rs=0:Everylistedboundaryvalueshouldinfluencethesolution:Forthe0Nvaluesfromuˆh,...,uˆhthisisthecaseiffAˆhsatisfiesthisproperty.Furthermore,the“local”indicesoftheequation,wheretheboundaryvaluesappear,arethesameforeachtimelevel.Forthevaluesfromu∈RM1theassertionfollowsfrom(7.100).0Fromtheconsiderationswehavethefollowingtheorem:Theorem7.28Considertheone-step-thetamethodintheform(7.66).Let(7.100)hold.IfthespatialdiscretizationAˆhsatisfies(1.32)(1),(2),(3)(i),and(5),thenacomparisonprincipleholds:n(1)Iffortwosetsofdatafi,u0ianduˆhi,n=0,...,Nandi=1,2,wehavef1((n−1+Θ)τ)≤f2((n−1+Θ)τ)forn=1,...,N,andnnu01≤u02;,uˆh1≤uˆh1forn=0,...,N,thennnuˆh1≤uˆh2forn=1,...,Nforthecorrespondingsolutions.nnIfuˆh1=uˆh2forn=1,...,N,thencondition(1.32)(5)canbeomitted. 7.5.MaximumPrinciplefortheOne-Step-ThetaMethod329(2)IfA˜additionallysatisfies(1.32)(6)∗,thenthefollowingweakhmaximumprincipleholds:nnmax(u˜h)r≤maxmax(u0)r,max(uˆh)r,r∈{1,...,M}r∈{1,...,M1}r∈{M1+1,...,M}n=0,...,Nn=0,...,Nwhereunu˜n:=h.huˆh(3)IfA˜hsatisfies(1.32)(1),(2),(3)(i),(4),(5),(6),(7),thenastrongmaximumprincipleinthefollowingsenseholds:nIfthecomponentsofu˜h,n=0,...,N,attainanonnegativemaxi-mumforsomespatialindexr∈{1,...,M1}andatsometimelevelk∈{1,...,N},thenallcomponentsforthetimelevelsn=0,...,kareequal.Proof:Onlypart(3)needsfurtherconsideration.Theorem1.9cannotbeapplieddirectlyto(7.97),sinceChisreducible.Therefore,theproofofTheorem1.9hastoberepeated:Weconcludethatthesolutionisconstantatallpointsthatareconnectedviaachainofneighbourstothepointwherethemaximumisattained.Accordingto(4)#theseincludeallgridpoints(x,tl)withx∈Ωhandl∈{0,...,k}.From(7.100)andthediscussionof(7)weseethattheconnectioncanalsobecontinuedtoboundaryvaluesuptolevelk.
Theadditionalcondition(7.100),whichmaybeweakenedtononstrictinequality,asseenabove,actuallyisatimesteprestriction:Consideragaintheexampleofthefive-pointstencildiscretizationoftheheatequationonarectangle,forwhichwehave(A)=4/h2.Thentheconditiontakeshjjtheformτ1<(7.105)h24(1−Θ)forΘ<1.Thisisverysimilartothecondition(7.87),(7.88)fortheexplicitEulermethod,butthebackgroundisdifferent.Asalreadynoted,theresultsabovealsoapplytothemoregeneralform(7.67)undertheassumption(7.69).Thecondition(7.100)thentakestheformτΘ(Ah)jj≤bjforj=1,...,M1.Exercises7.14Formulatetheresultsofthissection,inparticularcondition(7.100),fortheproblemintheform(7.67)withBhaccordingto(7.69)(i.e. 3307.DiscretizationofParabolicProblemsappropriateforfiniteelementdiscretizationswithmasslumping,see(7.74)).7.15Showthevalidityof(6)#fromExercise1.13forCifitholdsherehforAhandconcludeasinExercise1.13aweakmaximumprinciplefortheone-step-thetamethod.7.16Considertheinitial-boundaryvalueprobleminonespacedimensionut−εuxx+cux=fin(0,1)×(0,T),u(0,t)=g−(t),u(1,t)=g+(t),t∈(0,T),u(x,0)=u0(x),x∈(0,1),whereT>0andε>0areconstants,andc,f:(0,1)×(0,T)→R,u0:(0,1)→R,andg−,g+:(0,T)→Raresufficientlysmoothfunctionssuchthattheproblemhasaclassicalsolution.Defineh:=1/mandτ=T/Nforsomenumbersm,N∈N.Thentheso-calledfull-upwindfinitedifferencemethodforthisproblemreadsasfollows:Findasequenceofvectorsu0,...,uNbyhhun+1−unun+1−2un+1+un+1un+1−un+1un+1−un+1ii−εi+1ii−1−c−i+1i+c+ii−1τh2hhn+1=fi,i=1,...,m−1,n=0,...,N−1,wherec=c+−c−withc+=max{c,0},fn=f(ih,nτ),u0=u(ih),ii0un=g(nτ)andun=g(nτ).0−m+Provethataweakmaximumprincipleholdsforthismethod.7.6OrderofConvergenceEstimatesBasedonstabilityresultsalreadyderived,wewillinvestigatethe(orderof)convergencepropertiesoftheone-step-thetamethodfordifferentdis-cretizationapproaches.Althoughtheresultswillbecomparable,theywillbeindifferentnorms,appropriateforthespecificdiscretizationmethod,asalreadyseeninChapters1,3,and6.OrderofConvergenceEstimatesfortheFiniteDifferenceMethodFromSection1.4weknowthattheinvestigationofthe(orderof)convergenceofafinitedifferencemethodconsistsoftwoingredients:•(orderof)convergenceoftheconsistencyerror•stabilityestimates.ThelasttoolhasalreadybeenprovidedbyTheorem7.26andbyTheo-rem1.14,whichtogetherwiththeconsiderationsofSection7.5allowustoconcentrateontheconsistencyerror.Certainsmoothnesspropertieswillbe 7.6.OrderofConvergenceEstimates331requiredfortheclassicalsolutionuoftheinitialboundaryvalueproblem(7.1),whichinparticularmakesitsevaluationpossibleatthegridpointsxi∈Ωhateachinstanceoftimet∈[0,T]andalsoofvariousderivatives.Thevectorrepresentingthecorrespondinggridfunction(forafixedorder-ningofthegridpoints)willbedenotedbyU(t),orforshortbyU:=U(tn)fort=tn.Thecorrespondinggridpointsdependontheboundarycondi-tion.ForapureDirichletproblem,thegridpointswillbefromΩh,butifNeumannormixedboundaryconditionsappear,theyarefromtheenlargedsetΩ˜h:=Ωh∩(Ω∪Γ1∪Γ2).(7.106)Thentheerroratthegridpointsandeachtimelevelisgivenbyen:=Un−unforn=0,...,N,(7.107)hhwhereunisthesolutionoftheone-step-thetamethodaccordingto(7.66).hTheconsistencyerrorˆqhasagridfunctiononΩh×{t1,...,tN}orcorres-pondinglyasequenceofvectorsqˆninRM1forn=1,...,Nisthendefinedhby1n+1n+1nn+1qˆh:=U−U+ΘAhUτn+ΘAhU−qh((n+Θ)τ)(7.108)forn=0,...,N−1.Thentheerrorgridfunctionobviouslysatisfies1en+1−en+ΘAen+1+ΘAen=qˆn+1forn=0,...,N−1,τhhhhhhhe0=0(7.109)h(ornonvanishinginitialdataiftheinitialconditionisnotevaluatedexactlyatthegridpoints).Inthefollowingweestimatethegridfunctionˆqhinthediscretemaximumnormnqˆh∞:=max{|(qˆh)r||r∈{1,...,M1},n∈{1,...,N}}n=max{|qˆh|∞|n∈{1,...,N}},(7.110)i.e.,pointwiseinspaceandtime.AnalternativenormwouldbethediscreteL2-norm,i.e.,1/21/2NM1Nqˆ:=τhd|(qˆn)|2=τ|qˆn|2,(7.111)h0,hhrh0,hn=1r=1n=1usingthespatialdiscreteL2-normfrom(7.59),wherethesamenotationisemployed.IfforthesequenceofunderlyinggridpointsconsideredthereisaconstantC>0independentofthediscretizationparameterhsuchthat−dM1=M1(h)≤Ch,(7.112) 3327.DiscretizationofParabolicProblemsthenobviously,qˆ≤(CT)1/2qˆ,h0,hh∞sothattheL2-normisweakerthanthemaximumnorm.Condition(7.112)issatisfiedforsuchuniformgrids,asconsideredinSection1.2.Anorminbetweenisdefinedbynqˆh∞,0,h:=max{|qˆh|0,h|n=1,...,N},(7.113)whichisstrongerthan(7.111)andinthecaseof(7.112)weakerthanthemaximumnorm.nAnalogouslytoSection1.4,wedenoteUamendedbytheeliminatednnboundaryvaluesUˆ∈RM2bythevectorU˜∈RM.hForsimplicitywerestrictattention,atthebeginning,tothecaseofpureDirichletdata.Takingintoaccount(7.98)andassumingthatf((n−1+Θ)τ)isderivedfromthecontinuousright-handsidebyevaluationatthegridpoints,wegetn+11n+1ndqˆh=(U−U)−U(tn+Θτ)τdtn+1n+ΘA˜hU˜+ΘA˜hU˜−(LU)(tn+Θτ)=:S1+S2,(7.114)sothatS1,consistingofthefirsttwoterms,istheconsistencyerrorforthetimediscretization.HeredUandLUarethevectorsrepresentingthegridfunctionscorre-dtspondingtoduandLu,whichrequiresthecontinuityofthesefunctionsdtasinthenotionofaclassicalsolution.Wemakethefollowingassumption:Thespatialdiscretizationhastheorderofconsistencyαmeasuredin·∞(accordingto(1.17))ifthesolutionofthestationaryproblem(7.6)isinCp(Ω)forsomeα>0andp∈N.Forexample,fortheDirichletproblemofthePoissonequationandthefive-pointstencildiscretizationonarectangle,wehaveseeninChapter1thatα=2isvalidforp=4.Ifweassumeforu(·,t),ubeingthesolutionof(7.1),thatthespatialderivativesuptoorderpexistcontinuouslyandareboundeduniformlyint∈[0,T],(7.115)thenthereexistsaconstantC>0suchthat|(A˜U˜(t))−(Lu(·,t))(x)|≤Chα(7.116)hiiforeverygridpointxi∈Ωhandt∈[0,T].InthecaseofNeumannormixedboundaryconditions,thensomeoftheequationswillcorrespondtodiscretizationsoftheseboundaryconditions.Thisdiscretizationmaybedirectlyadiscretizationof(7.3)or(7.4)(typ-ically,ifone-sideddifferencequotientsareused)oralinearcombination 7.6.OrderofConvergenceEstimates333ofthediscretizationsofthedifferentialoperatoratxi∈Ω˜handoftheboundarydifferentialoperatorof(7.3)or(7.4)(toeliminate“artificial”gridpoints)(seeSection1.3).Thuswehavetotakexi∈Ω˜handinterpretLuin(7.116)asthismodifieddifferentialoperatorforxi∈Γ1∪Γ2justdescribedtoextendalltheabovereasoningtothegeneralcase.TheestimationofthecontributionS2onthebasisof(7.116)isdirectlypossibleforΘ=0orΘ=1,butrequiresfurthersmoothnessforΘ∈(0,1).WehaveS2=S3+S4,wheren+1nS3:=Θ(A˜hU˜−(LU)(tn+1))+Θ(A˜hU˜−(LU)(tn)),S4:=Θ(LU)(tn+1)+Θ(LU)(tn)−(LU)(tn+Θτ).ByTaylorexpansionweconcludeforafunctionv∈C2[0,T]that222Θ

1Θ

2Θv(tn+1)+Θv(tn)=v(tn+Θτ)+τΘv(tn)+Θv(tn)22forsomet1∈(t,t+Θτ),t2∈(t+Θτ,t),sothatnnnnnn+12|S4|∞≤Cτ(7.117)forsomeconstantC>0independentofτandhifforΘ∈(0,1)thesolutionuof(7.1)satisfies∂∂2Lu,2Lu∈C(QT).(7.118)∂t∂tThisisaquitesevereregularityassumption,whichoftendoesnothold.ForS3weconcludedirectlyfrom(7.116)that|S|≤Chα.(7.119)3∞ToestimateSwehavetodistinguishbetweenΘ=1andΘ=1:If122∂∂21∂3∂tu,∂t2u∈C(QT)andforΘ=2also∂t3u∈C(QT),(7.120)thenLemma1.2implies(forΘ=0,1,1,forΘ∈(0,1)againwithaTaylor2expansion)|S|≤Cτβ(7.121)1∞forsomeconstantC,independentofτandh,withβ=1forΘ=1and2β=2forΘ=1.2Thus,undertheadditionalregularityassumptions(7.115),(7.118),(7.120),andifthespatialdiscretizationhasorderofconsistencyαin 3347.DiscretizationofParabolicProblemsthemaximumnorm,i.e.,(7.116),thentheone-step-thetamethodhasthefollowingorderofconsistency:αβqˆh∞≤C(h+τ)(7.122)forsomeconstantC,independentofτandh,withβasin(7.121).Byusingaweakernormonemighthopetoachieveahigherorderofconvergence.Ifthisis,forexample,thecaseforthespatialdiscretization,e.g.,byconsideringthediscreteL2-norm·insteadof·,then0,h∞insteadof(7.116)wehaveA˜U˜(t)−Lu(·,t)≤Chα,(7.123)h0,hwherethetermsinthenormdenotethecorrespondinggridfunctions.Thenagainunder(weakerformsof)theadditionalregularityassump-tions(7.115),(7.118),(7.120)andassuming(7.112),wehaveqˆ≤C(hα+τβ).(7.124)h0,hBymeansofTheorem7.26wecanconcludethefirstorderofconvergenceresult:Theorem7.29Considertheone-step-thetamethodandassumethatthespatialdiscretizationmatrixAhhasabasisofeigenvectorswiwitheigen-valuesλi≥0,i=1,...,M1,orthogonalwithrespecttothescalarproduct·,·h,definedin(7.58).Thespatialdiscretizationhasorderofconsistencyαin·forsolutionsinCp(Ω).Ifτissuchthatthemethodissta-0,hbleaccordingto(7.95),thenforasufficientlysmoothsolutionuof(7.1)(e.g.,(7.115),(7.118),(7.120)),andforasequenceofgridpointssatisfying(7.112),themethodconvergesinthenorm·∞,0,hwiththeorderαβO(h+τ),whereβ=2forΘ=1andβ=1otherwise.2Proof:DuetoTheorem7.26and(7.109)wehavetoestimatetheNn1consistencyerrorinanormdefinedbyτn=1|qˆh|0,h(i.e.,adiscreteL-L2-norm),whichisweakerthanqˆ,inwhichtheestimatehasbeenh0,hverifiedin(7.124).
Againweseehereasmoothingeffectintime:TheconsistencyerrorhastobecontrolledonlyinadiscreteL1-sensetogainaconvergenceresultinadiscreteL∞-sense.Ifaconsistencyestimateisprovidedin·∞asin(7.122),aconvergenceestimatestillneedsthecorrespondingstability.InsteadofconstructingavectorasinTheorem1.14fortheformulation(7.97),wewillarguedirectlywiththehelpofthecomparisonprinciple(Theorem7.28,1)),whichwouldhavebeenpossiblealsoinSection1.4(seeExercise1.14). 7.6.OrderofConvergenceEstimates335Theorem7.30Considertheone-step-thetamethodandassumethatthespatialdiscretizationmatrixAhsatisfies(1.32)(1),(2),(3)(i)andassumeitsL∞-stabilitybytheexistenceofvectorsw∈RM1andaconstantC>0hindependentofhsuchthatAhwh≥1and|wh|∞≤C.(7.125)Thespatialdiscretizationhasorderofconsistencyαin·∞forsolutionsinCp(Ω).If(7.100)issatisfied,thenforasufficientlysmoothsolutionuof(7.1)(e.g.,(7.115),(7.118),(7.120))themethodconvergesinthenorm·∞withtheorderαβO(h+τ),whereβ=2forΘ=1andβ=1otherwise.2Proof:From(7.122)weconcludethat−Cˆ(hα+τβ)1≤qˆn≤Cˆ(hα+τβ)1forn=1,...,NhforsomeconstantCˆindependentofhandτ.Thus(7.109)implies1en+1−en+ΘAen+1+ΘAen≤Cˆ(hα+τβ)1,τhhhhhhe0=0.hSettingwn:=Cˆ(hα+τβ)wwithwfrom(7.125),thisconstantsequencehhhofvectorssatisfies1wn+1−wn+ΘAwn+1+ΘAwn≥Cˆ(hα+τβ)1.τhhhhhhTherefore,thecomparisonprinciple(Theorem7.28,(1))impliesen≤wn=Cˆ(hα+τβ)whhhforn=0,...,N,andanalogously,weseethatαβn−Cˆ(h+τ)wh≤eh,sothatnαβ(eh)j≤Cˆ(h+τ)(wh)j(7.126)foralln=0,...,Nandj=1,...,M1,andfinally,nαβαβ|eh|∞≤Cˆ(h+τ)|wh|∞≤Cˆ(h+τ)CwiththeconstantCfrom(7.125).
Notethatthepointwiseestimate(7.126)ismoreprecise,sinceitalsotakesintoaccounttheshapeofwh.Intheexampleofthefive-pointstencilwithDirichletconditionsontherectangle(seethediscussionaround(1.43)) 3367.DiscretizationofParabolicProblemstheerrorboundissmallerinthevicinityoftheboundary(whichistobeexpectedduetotheexactlyfulfilledboundaryconditions).OrderofConvergenceEstimatesfortheFiniteElementMethodWenowreturntotheone-step-thetamethodforthefiniteelementmethodasintroducedin(7.72).Inparticular,insteadofconsideringgridfunctionsasforthefinitedifferencemethod,thefiniteelementmethodallowsustoconsiderdirectlyafunctionUnfromthefinite-dimensionalapproximationspaceVhandthusfromtheunderlyingfunctionspaceV.Inthefollowing,anerroranalysisforthecaseΘ∈[1,1]undertheas-2sumptionu∈C2([0,T],V)willbegiven.Inanalogywiththedecompositionoftheerrorinthesemidiscretesituation,wewriteu(t)−Un=u(t)−Ru(t)+Ru(t)−Un=:(t)+θn.nnhnhnnThefirsttermoftheright-handsideistheerroroftheellipticprojectionatthetimetn,andforthistermanestimateisalreadyknown.Thefollowingidentityisusedtoestimatethesecondmemberoftheright-handside,whichimmediatelyresultsfromthedefinitionoftheellipticprojection:=>1n+1nn+1n(θ−θ),vh+a(Θθ+Θθ,vh)τ=0>1=((Rhu(tn+1)−Rhu(tn)),vh+a(ΘRhu(tn+1)+ΘRhu(tn),vh)τ=>01n+1nn+1n−(U−U),vh−a(ΘU+ΘU,vh)τ=0>1=(Rhu(tn+1)−Rhu(tn)),vh+a(Θu(tn+1)+Θu(tn),vh)τ0−bn+Θ(v)=h>1

=(Rhu(tn+1)−Rhu(tn)),vh−Θu(tn+1)+Θu(tn),vh0τ0=wn,v,h0wheren1

w:=(Rhu(tn+1)−Rhu(tn))−Θu(tn+1)−Θu(tn).τTakingintoconsiderationtheinequalitya(vh,vh)≥0,theparticularchoiceofthetestfunctionasv=Θθn+1+ΘθnyieldshΘθn+12+(1−2Θ)θn,θn+1−Θθn2≤τwn,Θθn+1+Θθn.0000ForΘ∈[1,1]wehave(1−2Θ)≤0,andhence2&'&'θn+1−θnΘθn+1+Θθn0000n+12nn+1n2=Θθ0+(1−2Θ)θ0θ0−Θθ0≤Θθn+12+(1−2Θ)θn,θn+1−Θθn2000&'nn+1n≤τw0Θθ0+Θθ0. 7.6.OrderofConvergenceEstimates337Dividingeachsidebytheexpressioninthesquarebrackets,wegetn+1nnθ0≤θ0+τw0.Therecursiveapplicationofthisinequalityleadstonθn+1≤θ0+τwj.(7.127)000j=0Thatis,itremainstoestimatethetermswj.Asimplealgebraic0manipulationyieldsn11w:=((Rh−I)u(tn+1)−(Rh−I)u(tn))+(u(tn+1)−u(tn))ττ

−Θu(tn+1)−Θu(tn).(7.128)Taylorexpansionwithintegralremainderimplies
tn+1


u(tn+1)=u(tn)+u(tn)τ+(tn+1−s)u(s)dstnand
tnu(t)=u(t)−u
(t)τ+(t−s)u

(s)ds.nn+1n+1ntn+1Usingtheaboverelationswegetthefollowingusefulrepresentationsofthedifferencequotientofuintn:
tn+11
1

(u(tn+1)−u(tn))=u(tn)+(tn+1−s)u(s)ds,ττtn
tn+11
1

(u(tn+1)−u(tn))=u(tn+1)+(tn−s)u(s)ds.ττtnMultiplyingthefirstequationbyΘandthesecondonebyΘ,thesummationoftheresultsyields1

(u(tn+1)−u(tn))=Θu(tn+1)+Θu(tn)τ
tn+11

+[Θtn+Θtn+1−s]u(s)ds.τtnSince|Θtn+Θtn+1−s|≤τ,thesecondterminthedecomposition(7.128)ofwncanbeestimatedas
1tn+1(u(t)−u(t))−Θu
(t)−Θu
(t)≤u

(s)ds.τn+1nn+1n00tnToestimatethefirsttermin(7.128),Taylorexpansionwithintegralremainderisappliedtothefunctionv(t):=(Rh−I)u(t).Thenwehave
tn+111
((Rh−I)u(tn+1)−(Rh−I)u(tn))=[(Rh−I)u(s)]ds.ττtn 3387.DiscretizationofParabolicProblemsWiththeassumptiononuusingthefactthatthederivativeandtheellipticprojectioncommute,weget
11tn+1((R−I)u(t)−(R−I)u(t))≤(R−I)u
(s)ds.τhn+1hnτh00tnWith(7.127)andsummingtheestimatesforwnweobtainthefollowing0result:Theorem7.31LetabeaV-elliptic,continuousbilinearform,u0h∈Vh,u∈V,Θ∈[1,1].Ifu∈C2([0,T],V),then02u(t)−Un≤u−Ru+(I−R)u(t)n00hh00hn0
tn
tn+(I−R)u
(s)ds+τu

(s)ds.h0000Remark7.32(i)Understrongersmoothnessassumptionsonuandbyde-tailedconsiderationsitcanalsobeshownthattheCrank–Nicolsonmethod(Θ=1)isoforder2inτ.2(ii)Contrarytothesemidiscretesituation(Theorem7.12),thefullydiscreteestimatedoesnotreflectanyexponentialdecayintime.UtilizingtheerrorestimatefortheellipticprojectionasinSection7.2(cf.Theorem7.12)andassumingu∈V∩H2(Ω),wehave05
tn6u(t)−Un≤u−u+Ch2u+u(t)+u
(s)dsn00h0002n22
0tn

+τu(s)0ds.0If,inaddition,u−u≤Ch2u,weobtain0h0002u(t)−Un≤C(u)(h2+τ),n0withC(u)>0dependingonthesolutionu(andthusonu0)butnotdependingonhandτ.ToconcludethissectionwegivewithoutproofasummaryoferrorestimatesforallpossiblevaluesofΘ:C(u)(h2+τ),ifΘ∈[1,1],2u(t)−Un≤C(u)(h2+τ2),ifΘ=1,(7.129)n02C(u)h2,ifΘ∈[0,1]andτ≤ϑh2,whereϑ>0isaconstantupperboundofthestepsizerelationτ/h2.Theoccurrenceofsucharestrictionisnotsurprising,sincesimilarrequirementshavealreadyappearedforthefinitedifferencemethod.Wealsomentionthattheaboverestrictiontoaconstantstepsizeτisonlyforsimplicityofthenotation.Ifavariablestepsizeτn+1isused(whichistypicallydeterminedbyastepsizecontrolstrategy),thenthenumberτinTheorem7.31istobereplacedbymaxn=0,...,N−1τn. 7.6.OrderofConvergenceEstimates339OrderofConvergenceEstimatesfortheFiniteVolumeMethodWenowconsidertheone-step-thetamethodforthefinitevolumemethodasintroducedin(7.75).Theerroranalysiswillruninasimilarwayasforthefiniteelementmethod.Wewritennnu(tn)−U=u(tn)−Rhu(tn)+Rhu(tn)−U=:(tn)+θ,whereRhistheauxiliaryoperatordefinedin(7.63).Soforthefirsttermoftheright-handside,anestimateisalreadyknown.Fromthedefinition(7.63)and(7.32),weimmediatelyderivethefollowingidentity:=>1n+1nn+1n(θ−θ),vh+ah(Θθ+Θθ,vh)τ=0,h>1=(Rhu(tn+1)−Rhu(tn)),vh+ah(ΘRhu(tn+1)+ΘRhu(tn),vh)τ=>0,h1n+1nn+1n−(U−U),vh−ah(ΘU+ΘU,vh)τ=0,h>1=(Rhu(tn+1)−Rhu(tn)),vh+a(Θu(tn+1)+Θu(tn),vh)τ0,h−fn+Θ,v=h0,h>1

=(Rhu(tn+1)−Rhu(tn)),vh−Θu(tn+1)+Θu(tn),vh0τ0,h+fn+Θ,v−fn+Θ,v=h0h0,h>1

=(Rhu(tn+1)−Rhu(tn)),vh−Θu(tn+1)+Θu(tn),vh0,hτ0,h+Θu
(t)+Θu
(t),v−Θu
(t)+Θu
(t),vn+1nh0,hn+1nh0+fn+Θ,v−fn+Θ,vh0h0,h=wn,v+rn(v),h0,hhwheren1

w:=(Rhu(tn+1)−Rhu(tn))−Θu(tn+1)−Θu(tn)τandn



r(vh):=Θu(tn+1)+Θu(tn),vh0,h−Θu(tn+1)+Θu(tn),vh0+fn+Θ,v−fn+Θ,v.h0h0,hUndertheassumptionsofTheorem6.15,weknowthatah(vh,vh)≥0forallv∈V.Theparticularchoiceofthetestfunctionasv=vΘ:=hhhhΘθn+1+Θθnyields,similarlytothefiniteelementcase,forΘ∈[1,1]the2 3407.DiscretizationofParabolicProblemsestimate&'&'θn+1−θnΘθn+1+Θθn0,h0,h0,h0,h≤τwn,vΘ+rn(vΘ)h0,hhrn(v)nhΘ≤τw0,h+supvh0,hvh∈Vhvh0,hrn(v)&'≤τwn+suphΘθn+1+Θθn.0,h0,h0,hvh∈Vhvh0,hDividingeachsidebytheexpressioninthesquarebrackets,wegetrn(v)θn+1≤θn+τwn+suph.0,h0,h0,hvh∈Vhvh0,hTherecursiveapplicationofthisinequalityleadstonnrj(v)θn+1≤θ0+τwj+τsuph.(7.130)0,h0,h0,hj=0j=0vh∈Vhvh0,hTherepresentationofwjobtainedinthesubsectiononthefiniteelementmethodyieldsthefollowingestimate:
tj+1
tj+1j1


w0,h≤(Rh−I)u(s)0,hds+u(s)0,hds.τtjtjFurthermore,byLemma7.14,wehave&'|rj(v)|≤ChΘ|u
(t)|+Θ|u
(t)|+|fj+Θ|v.hj+11,∞j1,∞1,∞h0,hUsingbothestimatesin(7.130),weobtainθn+10,h5

6tn+1tn+10


≤θ0,h+C(Rh−I)u(s)0,hds+τu(s)0,hds00An


+ChτΘ|u(0)|1,∞+|u(tj)|1,∞+Θ|u(tn+1)|1,∞j=1nB+|fj+Θ|1,∞j=05
6tn+1
tn+1≤θ0+C(R−I)u
(s)ds+τu

(s)ds0,hh0,h0,hA0B0+Chsup|u
(s)|+sup|f(s)|.1,∞1,∞s∈(0,tn+1)s∈(0,tn+1)Thisisthebasicestimate.Thefinalestimateiseasilyobtainedbythesameapproachasinthefiniteelementmethod.Insummary,wehavethefollowingresult.Theorem7.33InadditiontotheassumptionsofTheorem6.15,considerthefinitevolumemethodonDonalddiagrams.Furthermore,letu0h∈Vh, 7.6.OrderofConvergenceEstimates341u∈V∩H2(Ω),f∈C([0,T],C1(Ω)),Θ∈[1,1].Thenifu(t)issufficiently02smooth,thefollowingestimateisvalid:Anu(tn)−U0,h≤u0h−u00,h+Chu02+u(tn)2
tn+u
(s)ds+sup|u
(s)|21,∞0s∈(0,tn)B
tn

+sup|f(s)|1,∞+Cτu(s)0,hds.s∈(0,tn)0Exercise7.17VerifyRemark7.32. 8IterativeMethodsforNonlinearEquationsInthesamewayaslinear(initial-)boundaryvalueproblemsbythedis-cretizationtechniquesdiscussedinthisbookleadto(sequencesof)linearequations,wegetnonlinearequationsofsimilartypefromnonlinearprob-lems.Twoofthemwillbetreatedinthischapter.AsintheSections1.2,3.4,7.3,and6.2.4,wehavetoanswerthequestionofthequalityoftheapproximation,andasinSection2.5andChapter5,thequestionoftheapproximativeresolutionofthesystemsofequations.Wewillfocusonthelatterinthischapter.Ingeneral,theproblemmaybeformulatedindifferentequivalentsettings,namely:Findx∈Uwithf(x)=b.(8.1)Findx∈Uwithf(x)=0.(8.2)Thenxiscalledarootof(8.2)andazerooff.Findx∈Uwithf(x)=x.(8.3)Thenxiscalledafixedpoint.HereU⊂Rm,f:U→Rmisamapping,andb∈Rm.Thetransitionfromoneformulationtoanotherfollowsbyredefiningfinevidentways.Inmostcases,arootorafixedpointcannotbecalculated(withex-actarithmetic)inafinitenumberofoperations,butonlybyaniterativemethod,i.e.,byamappingΦ:U→U, 8.IterativeMethodsforNonlinearEquations343sothat(asin(5.7))forthesequencex(k+1):=Φx(k)(8.4)withgivenx(0)wegetx(k)→xfork→∞.(8.5)Herexisthesolutionof(8.1),(8.2),or(8.3).AswealreadystatedinSection5.1,inthecaseofacontinuousΦitfollowsfrom(8.4),(8.5)thatthelimitxsatisfiesx=Φ(x).(8.6)Thismeansthat(8.6)shouldimplythatxisasolutionof(8.1),(8.2),or(8.3).TheextensionofthedefinitionofconsistencyinSection5.1requirestheinverseimplication.Concerningtheerrorlevelthatweshouldachieveinrelationtotheap-proximationerrorofthediscretization,thestatementsintheintroductionofChapter5stillhold.Inadditiontothecriteriaofcomparisonforlin-earstationarymethodswenowhavetotakeintoaccountthefollowing:Methodsmay,iftheydoatall,convergeonlylocally,whichleadstothefollowingdefinition:Definition8.1Ifintheabovesituation(8.5)holdsforallx(0)∈U(i.e.,forarbitrarystartingvalues),then(x(k))iscalledgloballyconvergent.IfkanopenU˜⊂Uexistssuchthat(8.5)holdsforx(0)∈U˜,then(x(k))kiscalledlocallyconvergent.InthelattercaseU˜iscalledtherangeoftheiteration.Ontheotherhand,wemayobserveafasterconvergencethanthelinearconvergenceintroducedin(5.3):Definition8.2Let(x(k))beasequenceinRm,x∈Rm,and·anormkonRm.Thesequence(x(k))convergeslinearlytoxwithrespectto·ifkthereexistsaCwith01toxifkx(k)→xfork→∞andifthereexistsaC>0suchthatpx(k+1)−x≤Cx(k)−xforallk∈N.Thesequence(x(k))convergessuperlinearlytoxifkx(k+1)−xlim=0.k→∞x(k)−xThecasep=2isalsocalledquadraticconvergence.Thus,whilealinearlyconvergingmethodguaranteesareductionoftheerrorbyaconstantfactorC,thisreductionisimprovedstepbystepinthecaseofsuperlinearor 3448.IterativeMethodsforNonlinearEquationshigher-orderconvergence.Whenweencounterquadraticconvergence,forexample,thenumberofsignificantdigitsisdoubledineverystep(minusafixednumber),sothatusuallyonlyasmallnumberofiterationswillbenecessary.ForthisreasonvariantsofthequadraticallyconvergingNewtonmethod(Section8.2)areattractive.Buttherestrictionoflocalconvergencemayrequiremodificationstoenlargetherangeofconvergence.Toevaluatethecomplexityofanumericalmethodthenumberofele-mentaryoperationsforaniterationhastobeconsidered.Byanelementaryoperationwewantalsotounderstandtheevaluationoffunctionslikethesine,althoughthisismuchmorecostlythananordinaryfloating-pointoperation.Atypicalsubproblemduringaniterationcycleisthesolutionofasystemoflinearequations,analogouslytothesimplersystemsintheform(5.10)occurringinlinearstationaryproblems.Besidestheefforttoassemblethissystemofequations,wehavetoaccountfortheworktosolveit,whichcanbedonewithoneofthemethodsdescribedinSec-tion2.5andChapter5,i.e.,inparticular,againwithaniterativemethod.Wecallthisasecondaryorinneriteration,whichisattractivebecauseofthesparsestructureofthematricesoriginatingfromthediscretization,asalreadydiscussedinChapter5.Hereaninexactvariantmaybeuseful,withwhichtheinneriterationisperformedonlyuptoaprecisionthatconservestheconvergencepropertiesoftheouteriteration.Thenumericalcostfortheassemblingmay,infact,bemoreexpensivethanthecostforthein-neriteration.Hencemethodswithlowcostfortheassembling(butworseconvergence)shouldalsobeconsidered.Keepingthisinmind,wedevoteanintroductorychaptertothefixed-pointiterations,whichare,roughlyspeaking,methodsinwhichtheiterationΦcoincideswiththemappingf.8.1Fixed-PointIterationsForthefixed-pointformulation(8.3)thechoiceΦ:=fisevidentaccordingto(8.6);inotherwords,thefixed-pointiterationreadsx(k+1):=fx(k).(8.7)Todiminishthedistanceoftwosucceedingmembersofthesequence,i.e.,Φ(x(k+1))−Φ(x(k))=x(k+2)−x(k+1)<x(k+1)−x(k),itissufficientthattheiterationfunction(hereΦ=f)becontractive(seeAppendixA.4).Sufficientconditionsforacontractionaregivenbythefollowinglemma:Lemma8.3LetU⊂Rmbeopenandconvex,andg:U→Rmcontinuouslydifferentiable.IfsupDg(x)=:L<1x∈U 8.1.Fixed-PointIterations345holds,where·inRm,miscompatiblewith·inRm,thengiscontractinginU.Proof:Exercise8.1.
Therefore,ifU⊂Rmisopen,f:U⊂Rm→Rmiscontinuouslydifferentiable,andifthereexistssome˜x∈UwithDf(˜x)<1,thenthereexistsaclosedconvexneighbourhoodU˜of˜xwithDf(x)≤L<1forx∈U˜and,forexample,L=Df(˜x)+1(1−Df(˜x)),guaranteeingthe2contractivityoffinU.Theuniqueexistenceofafixedpointandtheconvergenceof(8.7)isguaranteedifthesetUwherefisacontractionismappedintoitself:Theorem8.4(Banach’sfixed-pointtheorem)LetU⊂Rm,U=∅,andUbeclosed.Letf:U→RmbecontractivewithLipschitzconstantL<1andf[U]⊂U.Thenwehave:(1)Thereexistsoneandonlyonefixedpointx∈Uoff.(2)Forarbitraryx(0)∈Uthefixedpointiteration(8.7)convergestox,andwehaveLx(k)−x≤x(k)−x(k−1)1−L(aposteriorierrorestimate)Lk≤x(1)−x(0)1−L(apriorierrorestimate).Proof:Thesequencex(k+1):=f(x(k))iswell-definedbecauseoff[U]⊂U.Weprovethat(x(k))isaCauchysequence(seeAppendixA.4).kx(k+1)−x(k)=f(x(k))−f(x(k−1))≤Lx(k)−x(k−1)≤L2x(k−1)−x(k−2)≤···≤Lkx(1)−x(0),(8.8)sothatforanyk,l∈Nx(k+l)−x(k)≤x(k+l)−x(k+l−1)+x(k+l−1)−x(k+l−2)+···+x(k+1)−x(k)≤(Lk+l−1+Lk+l−2+···+Lk)x(1)−x(0)=Lk(1+L+···+Ll−1)x(1)−x(0)∞≤LkLlx(1)−x(0)=Lk1x(1)−x(0).1−Ll=0 3468.IterativeMethodsforNonlinearEquationsThuswehavex(k+l)−x(k)→0fork→∞;i.e.,(x(k))kisaCauchysequenceandthusconvergestosomex∈RmbecauseofthecompletenessofRm.DuetotheclosednessofUweconcludethatx∈U.Sincewehavex(k+1)→x,fx(k)→f(x)fork→∞,xisalsoafixedpointoff.Thefixedpointisunique,becauseforfixedpointsx,x¯,x−x¯=f(x)−f(¯x)≤Lx−x¯,whichimmediatelyimpliesx=¯xbecauseofL<1.Moreover,wehavex(k)−x=f(x(k−1))−f(x)≤Lx(k−1)−x≤Lx(k−1)−x(k)+x(k)−x,andthusfrom(8.8),LLx(k)−x≤x(k)−x(k−1)≤Lk−1x(1)−x(0).1−L1−L
Remark8.5Thetheoremcanbegeneralized:Sinceweusedonlythecom-pletenessofRm,thepropositionholdseveninaBanachspace(X,·),whereU⊂Xisaclosedsubset.Thisenablesustodefineiterativeschemesdirectlyinthefunctionspacefornonlinearboundaryvalueproblems,whichmeansthattheresulting(linear)problemsintheiterationsteparetobediscretized.Soinsteadofproceedingintheorderdiscretization–iteration,wecanapplythesequenceiteration–discretization.Thisleadsingeneraltodifferentschemes,eveniftheapproacheshavebeenthesame.Wewillalwaysrefertothefirststrategy.AccordingtoLemma8.3wecanoftenconstructaclosedUsuchthatfiscontractiveonU.Itremainstoverifythatf[U]⊂U.Forthis,thefollowinglemmaishelpful:Lemma8.6LetU⊂Rm,f:U→Rm.Ifthereexistsay∈Uandar>0withBr(y)⊂U,withfcontractiveonBr(y)withLipschitzconstantL<1,sothaty−f(y)≤r(1−L),thenfhasoneandonlyonefixedpointinBr(y),and(8.7)converges.Proof:Exercise8.2.
8.1.Fixed-PointIterations347InthesettingofTheorem8.4thefixed-pointiterationisthusgloballyconvergentinU.InthesettingofLemma8.6itislocallyconvergentinU(globallyinBr(y)).WeseethatinthesituationofTheorem8.4thesequence(x(k))has,becauseof(k+1)(k)(k)x−x=f(x)−f(x)≤Lx−x,alinearorderofconvergence(andingeneralnotbetter).Asufficientconditionforlocalconvergenceofthecorrespondingorderisgivenbythefollowingtheorem:Theorem8.7LetU⊂Rmbeopen,Φ:U→Ucontinuous,thesequence(x(k))definedbyx(k+1):=Φx(k)foragivenx(0)∈U.Ifthereexistssomex¯∈U,anopenV⊂Uwithx¯∈V,andconstantsC,pwithp≥1,C≥0,andC<1forp=1,suchthatforallx∈V,pΦ(x)−x¯≤Cx−x¯holds,thentheiterationdefinedbyΦconvergeslocallytox¯oforderatleastp,andx¯isafixedpointofΦ.Proof:ChooseW=Br(¯x)⊂V,withr>0sufficientlysmall,suchthatW⊂VandCrp−1=:L<1.Ifx(k)∈W,thenweconcludebecauseofpx(k+1)−x¯=Φx(k)−x¯≤Cx(k)−x¯1,thentheiterationdefinedbyΦislocallyconvergenttox¯withorderofconvergencep,butnotbetter. 3488.IterativeMethodsforNonlinearEquationsProof:Taylor’sexpansionofΦat¯xgives,forx∈U,(p)Φ(ξ)pΦ(x)=Φ(¯x)+(x−x¯)withξ∈(x,x¯),p!andinthecasep=1wehave|Φ
(ξ)|<1forsufficientlysmall|x−x¯|.Thus,thereexistsaneighbourhoodVof¯xsuchthat|Φ(x)−x¯|≤C|x−x¯|pforallx∈VandC<1forp=1.Theorem8.7impliesorderofconvergencep.TheexampleΦ(x)=LxpwithL<1forp=1withthefixedpointx=0showsthatnoimprovementispossible.
Exercises8.1ProveLemma8.3withthehelpofthemeanvaluetheorem.8.2ProveLemma8.6.8.2Newton’sMethodandItsVariants8.2.1TheStandardFormofNewton’sMethodInthefollowingwewanttostudytheformulationstatedin(8.2),i.e.,theproblemoffindingthesolutionsoff(x)=0.ThesimplestmethodofChapter5,theRichardsoniteration(cf.(5.28)),suggeststhedirectapplicationofthefixed-pointiterationfor,e.g.,Φ(x):=−f(x)+x.Thisapproachsucceedsonlyif,inthecaseofadifferentiablef,theJacobianI−Df(x)issmallinthesenseofLemma8.3closetothesolution.HerewedenotebyDf(x)=(∂jfi(x))ijtheJacobiorfunctionalmatrixoff.Arelaxationmethodsimilarto(5.30)leadstothedampedvariants,whichwillbetreatedlater.Themethodinitscorrectorformulation,analogouslyto(5.10)withδ(k):=x(k+1)−x(k),is(k)(k)δ=−fx,(8.9)orinitsrelaxationformulationwithrelaxationparameterω>0,(k)(k)δ=−ωf(x).NowwewanttointroduceanotherapproachtodefineΦ:Letx(0)beanapproximationofazero.Animprovedapproximationisprobablygivenbythefollowing: 8.2.Newton’sMethodandVariants349•Replacefbyasimplefunctiongthatapproximatesfnearx(0)andwhosezeroistobedetermined.•Findx(1)asthesolutionofg(x)=0.Newton’smethodneedsthedifferentiabilityoff,andonechoosestheapproximatingaffine-linearfunctiongivenbyDf(x(0)),i.e.,g(x)=fx(0)+Dfx(0)x−x(0).UndertheassumptionthatDf(x(0))isnonsingular,thenewiteratex(1)isdeterminedbysolvingthesystemoflinearequationsDfx(0)x(1)−x(0)=−fx(0),(8.10)orformallyby(1)(0)(0)−1(0)x:=x−Dfxfx.Thissuggeststhefollowingdefinition:−1Φ(f)(x)=x−Df(x)f(x).(8.11)HereΦiswell-definedonlyifDf(x)isnonsingular.Thenx∈RmisazerooffifandonlyifxisafixedpointofΦ.Whenexecutingtheiteration,−1wedonotcalculateDfx(k)butonlythesystemofequationssimilarto(8.10).Thus,thekthiterationofNewton’smethodreadsasfollows:SolveDfx(k)δ(k)=−fx(k)(8.12)andsetx(k+1):=x(k)+δ(k).(8.13)Equation(8.13)hasthesameformas(5.10)withW=Df(x(k)),withtheresidualatx(k)(k)(k)d:=fx.Thusthesubproblemofthekthiterationiseasierinthesensethatitcon-sistsofasystemoflinearequations(withthesamestructureofdependenceasf;seeExercise8.6).Inthesamesensethesystemofequations(5.10)inthecaseoflinearstationarymethodsis“easier”tosolvethantheorig-inalproblemofthesametype.Furthermore,Wisingeneraldifferentfordifferentk.Anapplicationof(8.12),(8.13)toAx=b,i.e.,Df(x)=Aforallx∈Rmresultsin(5.10)withW=A,amethodconverginginonestep,whichjustreformulatestheoriginalproblem:(0)(0)Ax−x=−Ax−b. 3508.IterativeMethodsforNonlinearEquationsTherangeoftheiterationmaybeverysmall,ascanbeshownalreadybyone-dimensionalexamples.Butinthisneighbourhoodofthesolutionwehave,e.g.,form=1,thefollowing:Corollary8.9Letf∈C3(R)andletx¯beasimplezerooff(i.e.,f
(¯x)=0).ThenNewton’smethodconvergeslocallytox¯,oforderatleast2.Proof:ThereexistsanopenneighbourhoodVof¯xsuchthatf
(x)=0forallx∈V;i.e.,Φiswell-definedby(8.11),continuousonV,and¯xisafixedpointofΦ.AccordingtoCorollary8.8itsufficestoshowthatΦ
(¯x)=0:f
(x)2−f(x)f

(x)f

(x)
Φ(x)=1−=f(x)=0forx=¯x,f
(x)2f
(x)2andΦ

existscontinuously,becausef∈C3(R).
Inthefollowingwewanttodevelopagenerallocaltheoremofconver-genceforNewton’smethod(accordingtoL.V.Kantorovich).ItnecessitatesonlytheLipschitzcontinuityofDfandensurestheexistenceofazero,too.HerewealwayssupposeafixednormonRmandconsideracompatiblenormonRm,m.Asaprerequisiteweneedthefollowinglemma:Lemma8.10LetC⊂Rmbeconvex,open,f:C→Rmdifferentiable,00andsupposethereexistsγ>0suchthatDf(x)−Df(y)≤γx−yforallx,y∈C0.(8.14)Thenforallx,y∈C0,γ2f(x)−f(y)−Df(y)(x−y)≤x−y.2Proof:Letϕ:[0,1]→Rmbedefinedbyϕ(t):=f(y+t(x−y)),forarbitrary,fixedx,y∈C0.Thenϕisdifferentiableon[0,1]andϕ
(t)=Df(y+t(x−y))(x−y).Thusfort∈[0,1]wehaveϕ
(t)−ϕ
(0)=(Df(y+t(x−y))−Df(y))(x−y)2≤Df(y+t(x−y))−Df(y)x−y≤γtx−y.For
1∆:=f(x)−f(y)−Df(y)(x−y)=ϕ(1)−ϕ(0)−ϕ
(0)=(ϕ
(t)−ϕ
(0))dt0wealsoget
1
1

212∆≤ϕ(t)−ϕ(0)dt≤γx−ytdt=γx−y.002
8.2.Newton’sMethodandVariants351Nowweareabletoconcludelocal,quadraticconvergence:Theorem8.11LetC⊂Rmbeconvex,openandf:C→Rmdifferen-tiable.Forx(0)∈Cthereexistα,β,γ>0suchthath:=αβγ/2<1,r:=α/(1−h),B¯x(0)⊂C.rFurthermore,werequire:(i)DfisLipschitzcontinuousonC=Bx(0)forsomeε>0with0r+εconstantγinthesenseof(8.14).(ii)Forallx∈Bx(0)thereexistsDf(x)−1andDf(x)−1≤β.r−1(iii)Dfx(0)fx(0)≤α.Then:(1)TheNewtoniteration(k+1)(k)(k)−1(k)x:=x−Dfxfxiswell-definedandx(k)∈Bx(0)forallk∈N.r(2)x(k)→x¯fork→∞andf(¯x)=0.kβγ2h2−1(3)x(k+1)−x¯≤x(k)−x¯andx(k)−x¯≤α21−h2kfork∈N.Proof:(1):Toshowthatx(k+1)iswell-defineditissufficienttoverifyx(k)∈Bx(0)(⊂C)forallk∈N.rByinductionweprovetheextendedpropositionk−1x(k)∈Bx(0)andx(k)−x(k−1)≤αh2−1forallk∈N.(8.15)rTheproposition(8.15)holdsfork=1,becauseaccordingto(iii),x(1)(0)=Df(0)−1(0)≤α0thereexistsaδ>0suchthatforx,y∈Bδ(¯x),f(y)x−x¯≤(1+)κ(Df(¯x))f(x)y−x¯.Proof:See[22,p.69,p.72]andExercise8.4.
Hereκistheconditionnumberinamatrixnormthatisconsistentwiththechosenvectornorm.Forx=x(k)andy=x(0)weget(locally)thegeneralizationof(5.16).8.2.2ModificationsofNewton’sMethodModificationsofNewton’smethodaimintwodirections:•Reductionofthecostoftheassemblingandthesolutionofthesys-temofequations(8.12)(withoutasignificantdeteriorationofthepropertiesofconvergence).•Enlargementoftherangeofconvergence.Wecanaccountforthefirstaspectbysimplifyingthematrixin(8.12)(modifiedorsimplifiedNewton’smethod).Theextremecaseisthereplace-mentofDfx(k)bytheidentitymatrix;thisleadsustothefixed-pointiteration(8.9).Ifthemappingfconsistsofanonlinearandalinearpart,f(x):=Ax+g(x)=0,(8.18)thenthesystemofequations(8.12)oftheNewtoniterationreadsasA+Dgx(k)δ(k)=−fx(k).Astraightforwardsimplificationinthiscaseisthefixed-pointiterationAδ(k)=−fx(k).(8.19)Itmaybeinterpretedasthefixed-pointiteration(8.9)ofthesystemthatispreconditionedwithA,i.e.,ofA−1f(x)=0.In(8.19)thematrixisidenticalineveryiterationstep;therefore,ithastobeassembledonlyonce,andifweuseadirectmethod(cf.Section2.5),theLUfactorizationhastobecarriedoutonlyonce.Thuswithforward 3548.IterativeMethodsforNonlinearEquationsandbackwardsubstitutionwehaveonlytoperformmethodswithlowercomputationalcost.Foriterativemethodswecannotrelyonthisadvantage,butwecanexpectthatx(k+1)isclosetox(k),andconsequentlyδ(k,0)=0constitutesagoodinitialguess.Accordingly,theassemblingofthematrixgetsmoreimportantwithrespecttotheoverallcomputationalcost,andsavingsduringtheassemblingbecomerelevant.Wegetasystemofequationssimilarto(8.19)byapplyingthechordmethod(seeExercise8.3),wherethelinearapproximationoftheinitialiterateismaintained,i.e.,Dfx(0)δ(k)=−fx(k).(8.20)IfthematrixBx(k),whichapproximatesDfx(k),ischangingineachiterationstep,i.e.,(k)(k)(k)Bxδ=−fx,(8.21)thentheonlyadvantagecanbeapossiblyeasierassemblingorsolvabilityofthesystemofequations.Ifthepartialderivatives∂jfi(x)aremoredifficulttoevaluatethanthefunctionfi(y)itself(orpossiblynotevaluableatall),thentheapproximationofDf(x(k))bydifferencequotientscanbetakenintoconsideration.Thiscorrespondsto1Bx(k)e=f(x+he)−f(x)(8.22)jjhforcolumnjofBx(k)withafixedh>0.Thenumberofcomputationsfortheassemblingofthematrixremainsthesame:m2forthefullmatrixandanalogouslyforthesparsematrix(seeExercise8.6).Observethatnumericaldifferentiationisanill-posedproblem,whichmeansthatweshouldideallychooseh∼δ1/2,whereδ>0istheerrorlevelintheevaluationoff.EventhenwecanmerelyexpectDfx(k)−Bx(k)≤Cδ1/2(see[22,pp.80f.]).Thusinthebestcasewecanexpectonlyhalfofthesignificantdigitsofthemachineprecision.Thesecondaspectoffacilitatedsolvabilityof(8.21)occursifthereappear“small”entriesintheJaco-bian,duetoaproblem-dependentweakcouplingofthecomponents,andtheseentriesmaybeskipped.Take,forexample,aDfx(k)withablockstructureasin(5.39):(k)mi,mjDfx=Aij,Aij∈R,ijsuchthattheblocksAijmaybeneglectedforj>i.Thenthereresultsanestedsystemofequationsofthedimensionsm1,m2,...,mp.ThepossibleadvantagesofsuchsimplifiedNewton’smethodshavetobeweightedagainstthedisadvantageofadeteriorationintheorderofconvergence:InsteadofanestimationlikethatinTheorem8.11,(3),we 8.2.Newton’sMethodandVariants355havetoexpectanadditionaltermBx(k)−Dfx(k)x(k)−x.Thismeansonlylinearor—bysuccessiveimprovementoftheapproxi-mation—superlinearconvergence(see[22,pp.75ff.]).Ifwehaveagoodinitialiterate,itmayoftenbeadvantageoustoperformasmallnumberofstepsofNewton’smethod.SointhefollowingwewilltreatagainNewton’smethod,althoughthesubsequentconsiderationscanalsobetransferredtoitsmodifications.Ifthelinearproblems(8.12)aresolvedwithaniterativescheme,wehavethepossibilitytoadjusttheaccuracyofthealgorithminordertoreducethenumberofinneriterations,withouta(severe)deteriorationoftheconvergenceoftheouteriterationoftheNewtoniteration.SodealingwithsuchinexactNewton’smethods,wedetermineinsteadofδ(k)from(8.12)onlyδ˜(k),whichfulfils(8.12)onlyuptoaninnerresidualr(k),i.e.,Dfx(k)δ˜(k)=−fx(k)+r(k).Thenewiterateisgivenbyx(k+1):=x(k)+δ˜(k).Theaccuracyofδ˜(k)isestimatedbytherequirementr(k)≤ηfx(k)(8.23)kwithcertainpropertiesforthesequence(ηk)kthatstillhavetobede-termined.Sincethenaturalchoiceoftheinitialiterateforsolving(8.12)isδ(k,0)=0,(8.23)correspondstotheterminationcriterion(5.15).Conditionsforηkcanbededucedfromthefollowingtheorem:Theorem8.13Let(8.17)holdandconsidercompatiblematrixandvectornorms.Thenthereexistsforevery>0aδ>0suchthatforx(k)∈B(¯x),δ−1x(k+1)−x¯≤x(k)−Dfx(k)fx(k)−x¯(8.24)+(1+)κDf(¯x)ηkx(k)−x¯.Proof:BythechoiceofδwecanensurethenonsingularityofDf(x(k)).Fromδ˜(k)(k)−1(k)(k)−1(k)=−Dfxfx+Dfxritfollowsthatx(k+1)−x¯=x(k)−x¯+δ˜(k)x(k)(k)−1(k)+Df(k)−1(k).≤−x¯−DfxfxxrTheassertioncanbededucedfromtheestimationDf(k)−1(k)≤(1+)1/2Df(¯x)−1r(k)xr 3568.IterativeMethodsforNonlinearEquations≤(1+)1/2Df(¯x)−1η(1+)1/2Df(¯x)x(k)−x¯.kHereweusedExercise8.4(2),(3)and(8.23).
Thefirstpartoftheapproximationcorrespondstotheerroroftheex-actNewtonstep,whichcanbeestimatedusingthesameargumentasinTheorem8.11,(3)(withExercise8.4,(2))byx(k)(k)−1(k)≤(1+)1/2Df(¯x)−1γx(k)2−Dfxfx−x¯−x¯.2Thisimpliesthefollowingresult:Corollary8.14LettheassumptionsofTheorem8.13besatisfied.Thenthereexistδ>0andη>¯0suchthatforx(0)∈B(¯x)andη≤η¯forallδkk∈NfortheinexactNewton’smethodthefollowinghold:(1)Thesequencex(k)convergeslinearlytox¯.k(2)Ifη→0fork→∞,thenx(k)convergessuperlinearly.kk(3)Ifη≤Kfx(k)foraK>0,thenx(k)convergesquadratical-kkly.Proof:Exercise8.5.
Theestimation(8.24)suggeststhatwecarefullychooseaveryfinelevelofaccuracy¯ηoftheinneriterationtoguaranteetheabovestatementsofconvergence.Thisisparticularlytrueforill-conditionedDf(¯x)(whichiscommonfordiscretizationmatrices:See(5.34)).Infact,theanalysisintheweightednorm·=Df(¯x)·showsthatonlyηk≤η<¯1hastobeensured(cf.[22,pp.97ff.]).Withthisandonthebasisoff(k)2f(k−1)2η˜k=αx/xforsomeα≤1wecanconstructηkinanadaptiveway(see[22,p.105]).MostoftheiterativemethodsintroducedinChapter5donotrequiretheexplicitknowledgeofthematrixDfx(k).ItsufficesthattheoperationDfx(k)ybefeasibleforvectorsy,ingeneralforfewerthanmofthem;i.e.,thedirectionalderivativeoffinx(k)indirectionyisneeded.Thusincaseadifferenceschemeforthederivativesoffshouldbenecessaryorreasonable,itismoreconvenienttochoosedirectlyadifferenceschemeforthedirectionalderivative.SincewecannotexpectconvergenceofNewton’smethodingeneral,werequireindicatorsfortheconvergencebehaviouroftheiteration.Thesolution¯xisinparticularalsothesolutionof2mMinimizef(x)forx∈R. 8.2.Newton’sMethodandVariants357Letx(0),τ>0,η,Θ¯∈(0,1),k=0,i=0begiven.0(1)δ˜(k,0):=0,i:=1.(2)Determinetheithiterateδ˜(k,i)forDf(x(k))δ˜(k)=−f(x(k))andcalculater(i):=Df(x(k))δ˜(k,i)+f(x(k)).(3)Ifr(i)≤ηf(x(k)),thengoto(4),kelseseti:=i+1andgoto(2).(4)δ˜(k):=δ˜(k,i).(5)x(k+1):=x(k)+δ˜(k).(6)Iff(x(k+1))>Θf(x(k)),interrupt.(7)Iff(x(k+1))≤τf(x(0)),end.Elsecalculateηk+1,setk:=k+1,andgoto(1).Table8.1.InexactNewton’smethodwithmonotonicitytest.Thuswecouldexpectadescentofthesequenceofiterates(x(k))inthisfunctional,i.e.,f(x(k+1))≤Θ¯f(x(k))foraΘ¯<1.Ifthismonotonicitytestisnotfulfilled,theiterationisterminated.SuchanexampleofaninexactNewton’smethodisgiveninTable8.1.Inordertoavoidtheterminationofthemethodduetodivergence,thecontinuationmethodshavebeendeveloped.Theyattributetheproblemf(x)=0toafamilyofproblemstoprovidesuccessivelygoodinitialiter-ates.TheapproachpresentedattheendofSection8.3fortime-dependentproblemsissimilartothecontinuationmethods.Anotherapproach(whichcanbecombinedwiththelatter)modifiesthe(inexact)Newton’smethod,sothattherangeofconvergenceisenlarged:Applyingthedamped(inex-act)Newton’smethodmeansreducingthesteplengthofx(k)tox(k+1)aslongasweobserveadecreaseconformabletothemonotonicitytest.Onestrategyofdamping,termedArmijo’srule,isdescribedinTable8.2andreplacesthesteps(1),(5),and(6)inTable8.1.ThusdampingNewton’smethodmeansalsoarelaxationsimilarto(5.30),whereω=λkisbeingadjustedtotheiterationstepasin(5.41).IntheformulationofTable8.2theiterationmayeventuallynotterminateifin(5)λkissuccessivelyreduced.Thismustbeavoidedinapracticalimplementationofthemethod.Butexceptforsituationswheredivergenceisobvious,thissituationwillnotappear,becausewehavethefollowingtheorem: 3588.IterativeMethodsforNonlinearEquationsLetadditionallyα,β∈(0,1)begiven.(1)δ˜(k,0):=0,i:=1,λ:=1.k(5)Iff(x(k)+λδ˜(k))≥(1−αλ)f(x(k)),setλ:=βλkkkkandgoto(5).(6)x(k+1):=x(k)+λδ˜(k).kTable8.2.DampedinexactNewtonstepaccordingtoArmijo’srule.Theorem8.15Letα,β,γ>0existsuchthatconditions(i),(ii)ofTheo-)rem8.11onB(x(k))holdforthesequence(x(k))definedaccordingk∈NrktoTable8.2.Letηk≤η¯foranη<¯1−α.Theniffx(0)=0,thereexistsaλ>¯0suchthatλ≥λ¯forallk∈N.Ifkfurthermorex(k)isbounded,thenthereexistsazerox¯,satisfying(8.17)kandx(k)→x¯fork→∞.Thereexistsak0∈Nsuchthatfork≥k0therelationλk=1holds.Proof:See[22,pp.139ff.].
Weseethatinthefinalstageoftheiterationweagaindealwiththe(inexact)Newton’smethodwiththepreviouslydescribedbehaviourofconvergence.Finally,thefollowingshouldbementioned:Theproblemf(x)=0andNewton’smethodareaffine-invariantinthesensethatatransitiontoAf(x)=0withanonsingularA∈Rm,mchangesneithertheproblemnortheiterationmethod,sinceD(Af)(x)−1Af(x)=Df(x)−1f(x).AmongtheassumptionsofTheorem8.11,(8.14)isnotaffine-invariant.ApossiblealternativewouldbeDf(y)−1(Df(x)−Df(y))≤γx−y,whichfulfilstherequirement.WiththeproofofLemma8.10itfollowsthat−1γ2Df(y)(f(x)−f(y)−Df(y)(x−y))≤x−y.2WiththisargumentasimilarvariantofTheorem8.11canbeproven. 8.2.Newton’sMethodandVariants359Thetestofmonotonicityisnotaffine-invariant,soprobablythenaturaltestofmonotonicityDf(k)−1(k+1)≤Θ¯Df(k)−1(k)xfxxfxhastobepreferred.Thevectorontheright-handsidehasalreadybeencalculated,being,exceptforthesign,theNewtoncorrectionδ(k).Butforthevectorintheleft-handside,−δ¯(k+1),thesystemofequationsDfx(k)δ¯(k+1)=−fx(k+1)additionallyhastoberesolved.Exercises8.3Considerthechordmethodasdescribedin(8.20).Provetheconvergenceofthismethodtothesolution¯xunderthefollowingassumptions:(1)Let(8.17)withBr(¯x)⊂Chold,&'−1(2)Df(x(0))≤β,(3)2βγr<1,(4)x(0)∈B(¯x).r8.4Letassumption(8.17)hold.Proveforcompatiblematrixandvectornormsthatforevery>0thereexistsaδ>0suchthatforeveryx∈Bδ(¯x),(1)Df(x)≤(1+)1/2Df(¯x),(2)Df(x)−1≤(1+)1/2Df(¯x)−1(employ(I−M)−1≤1/(1−M)forM<1),(3)(1+)−1/2Df(¯x)−1−1x−x¯≤f(x)≤(1+)1/2Df(¯x)x−x¯,(4)Theorem8.12.8.5ProveCorollary8.14.8.6LetU⊂Rmbeopenandconvex.Considerproblem(8.2)withcon-tinuouslydifferentiablef:U→Rm.Fori=1,...,mletJ⊂{1,...,m}ibedefinedby∂jfi(x)=0forj/∈Jiandeveryx∈U. 3608.IterativeMethodsforNonlinearEquationsThentheoperatorfissparselyoccupiedifli:=|Ji|0,holds:Thereexistssomeα≥0suchthatψ
(u)≥αforallu∈R.(8.28)Moreprecisely,wehaveforη∈RM,if(8.28)isvalid,
ηTDG(ξ¯)η=ψ
(¯u)|Pη|2dx≥αPη2.0ΩForsuchamonotonenonlinearitythepropertiesofdefinitenessofthestiff-nessmatrixSmaybe“enforced”.If,ontheotherhand,wewanttomakeuseofthepropertiesofanM-matrixthatcanbeensuredbytheconditions∗(1.32)or(1.32),thenitisnotclearwhetherthesepropertiesareconservedafteradditionofDG(ξ¯).ThisisduetothefactthatDG(ξ¯)isasparsema-trixofthesamestructureasS,butitalsoentailsaspatialcouplingthatisnotcontainedinthecontinuousformulation(8.25).NumericalQuadratureOwingtotheabovereason,theuseofanode-orientedquadraturerulefortheapproximationofG(ξ)issuggested,i.e.,aquadratureformulaofthetypeMQ(f):=ωif(ai)forf∈C(Ω)¯(8.29)i=1withweightsωi∈R.Suchaquadratureformularesultsfrom
Q(f):=I(f)dxforf∈C(Ω)¯,(8.30)ΩwhereMI:C(Ω)¯→Vh,I(f):=f(ai)ϕi,i=1istheinterpolationoperatorofthedegreesoffreedom.Forthisconsidera-tionwethusassumethatonlyLagrangianelementsenterthedefinitionof 3628.IterativeMethodsforNonlinearEquationsVh.Inthecaseof(8.30)theweightsin(8.29)arehencegivenby
ωi=ϕidx.ΩThiscorrespondstothelocaldescription(3.116).Morespecifically,weget,forexample,forthelinearapproachonsimplicesasageneralizationofthecompositetrapezoidalrule,1ωi=|K|,(8.31)d+1K∈Thwithai∈KwithddenotingthespatialdimensionandThtheunderlyingtriangulation.ApproximationofthemappingGbyaquadratureruleofthetype(8.29)givesG˜(ξ)=G˜j(ξ)withG˜j(ξ)=ωjψ(ξj),jbecauseofϕj(ai)=δij.WeseethattheapproximationG˜hasthepropertythatG˜jdependsonlyonξj.WecallsuchaG˜adiagonalfield.Qualitatively,thiscorrespondsbettertothecontinuousformulation(8.25)andleadstothefactthatDG˜(ξ¯)isdiagonal:DG˜(ξ¯)=ωψ
(ξ¯)δ.(8.32)ijjjijIfweimposethatallquadratureweightsωiarepositive,whichisthecasein(8.31)andalsoinotherexamplesinSection3.5.2,alloftheabovecon-siderationsaboutthepropertiesofDG˜(ξ¯)andS+DG˜(ξ¯)remainvalid;∗additionally,ifSisanM-matrix,becausetheconditions(1.32)or(1.32)arefulfilled,thenS+DG˜(ξ¯)remainsanM-matrix,too.Thisisjustifiedbythefollowingfact(compare[34]and[5];cf.(1.33)forthenotation):IfAisanM-matrixandB≥Awithbij≤0fori=j,(8.33)thenBisanM-matrixaswell.ConditionsofConvergenceComparingtherequirementsforthefixed-pointiterationandNewton’smethodstatedinthe(convergence)Theorems8.4and8.11,weobservethattheconditionsinTheorem8.4canbefulfilledonlyinspecialcases,whereS−1DG˜(ξ¯)issmallaccordingtoasuitablematrixnorm(seeLemma8.3).Butitisalsodifficulttodrawgeneralconclusionsaboutrequirement(iii)inTheorem8.11,whichtogetherwithh<1quantifiestheclosenessoftheinitialiteratetothesolution.Thepostulation(i),ontheotherhand,ismetfor(8.27)and(8.32)ifψ
isLipschitzcontinuous(seeExercise8.7).Concerningthepostulation(ii)wehavethefollowing:Letψbemonotonenondecreasing(i.e.,(8.28)holdswithα≥0)andletSbesymmetricandpositivedefinite,whichistrueforaproblemwithoutconvectionterms 8.3.SemilinearEllipticandParabolicProblems363(compare(3.27)).ThenwehaveinthespectralnormS−1=1/λ(S).2minHereλmin(S)>0denotesthesmallesteigenvalueofS.Hence(S+DG(ξ))−1=1/λS+DG(ξ¯)≤1/λ(S)=S−1,2minmin2andconsequently,requirement(ii)isvalidwithβ=S−1.2Concerningthechoiceoftheinitialiterate,thereisnogenerallysuccessfulstrategy.Wemaychoosethesolutionofthelinearsubproblem,i.e.,(0)Sξ=b.(8.34)Shoulditfailtoconvergeevenwithdamping,thenwemayapply,asageneralizationof(8.34),thecontinuationmethodtothefamilyofproblemsf(λ,ξ):=S+λG(ξ)−b=0withcontinuationparameterλ∈[0,1].Ifalltheseproblemshavesolutionsξ=ξλsothatDf(ξ;λ)existsandisnonsingularinaneighbourhoodofξλ,andifthereexistsacontinuoussolutiontrajectorywithoutbifurcation,then[0,1]canbediscretizedby0=λ0<λ1<···<λN=1,andsolutionsξλioff(ξ;λi)=0canbeobtainedbyperformingaNewtoniterationwiththe(approximative)solutionforλ=λi−1asstartingiterate.Sincetheξλifori0,considertheboundaryvalueproblem(−ku
+u)
=0inΩ:=(0,1),u(0)=u(1)−1=0.Itssolutionis1−exp(x/k)u(x)=.1−exp(1/k)Aroughsketchofthegraph(Figure9.1)showsthatthisfunctionhasasignificantboundarylayerattherightboundaryoftheintervalevenforthecomparativelysmallglobalP´ecletnumberPe=100.Inthelargersubinterval(about(0,0.95))itisverysmooth(nearlyconstant),whereasintheremainingsmallsubinterval(about(0.95,1))theabsolutevalueofitsfirstderivativeislarge.Givenanequidistantgridofwidthh=1/(M+1),M∈N,adis-cretizationbymeansofsymmetricdifferencequotientsyieldsthedifference 3709.DiscretizationofConvection-DominatedProblems10.80.60.40.2000.20.40.60.81Figure9.1.Solutionfork=0.01.equationsui−1−2ui+ui+1ui+1−ui−1−k+=0,i∈{1,...,M}=:Λ,h22hu0=uM+1−1=0.Collectingthecoefficientsandmultiplyingtheresultby2h,wearriveat2k4k2k−−1ui−1+ui+−+1ui+1=0,i∈Λ.hhhIfwemaketheansatzu=λi,thedifferenceequationscanbesolvediexactly:i1−2k+h2k−hui=M+1.1−2k+h2k−hInthecase2k0,c∈C1(Ω,Rd),r∈C(Ω),f∈L2(Ω).Furthermore,assumethatthefollowinginequalityisvalidinΩ,wherer>0isaconstant:r−1∇·c≥r.020Thenthebilinearforma:V×V→R,V:=H1(Ω),correspondingto0theboundaryvalueproblem(9.2),readsas(cf.(3.23))
a(u,v):=[ε∇u·∇v+c·∇uv+ruv]dx,u,v∈V.(9.9)ΩTogetanellipticityestimateofa,wesetu=v∈Vin(9.9)andtaketherelation2v(c·∇v)=c·∇v2intoaccount.Then,bypartialintegrationofthemiddleterm,weobtain2a(v,v)=ε|v|1+c·∇v,v0+rv,v0 3749.DiscretizationofConvection-DominatedProblems=>=>212212=ε|v|1−∇·c,v+rv,v0=ε|v|1+r−∇·c,v.2200Introducingtheso-calledε-weightedH1-normby221/2vε:=ε|v|1+v0,(9.10)weimmediatelyarriveattheestimatea(v,v)≥ε|v|2+rv2≥α˜v2,(9.11)100εwhere˜α:=min{1,r0}doesnotdependonε.Duetoc·∇u=∇·(cu)−(∇·c)u,partialintegrationyieldsforarbitraryu,v∈Vtheidentityc·∇u,v=−u,c·∇v−(∇·c)u,v.000Sowegetthecontinuityestimate|a(u,v)|≤ε|u|1|v|1+c0,∞u0|v|1+(|c|1,∞+r0,∞)u0v0√√≤(ε|u|1+u0){(ε+c0,∞)|v|1+(|c|1,∞+r0,∞)v0}≤M˜uεv1,(9.12)√whereM˜:=2max{ε+c0,∞,|c|1,∞+r0,∞}.Sinceweareinterestedinthecaseofsmalldiffusionε>0andpresentconvection(i.e.,c0,∞>0),thecontinuityconstantM˜canbeboundedindependentofε.Itisnotverysurprisingthattheobtainedcontinuityestimateisnonsymmetric,sincealsothedifferentialexpressionLbehaveslikethat.Passingovertoasymmetricestimateresultsinthefollowingrelation:M˜|a(u,v)|≤√uεvε.εNow,ifVh⊂Vdenotesafiniteelementspace,wecanargueasintheproofofC´ea’slemma(Theorem2.17)andgetanerrorestimateforthecorre-spondingfiniteelementsolutionuh∈Vh.Todothis,thenonsymmetriccontinuityestimate(9.12)issufficient.Indeed,forarbitraryvh∈Vh,wehaveα˜u−u2≤a(u−u,u−u)=a(u−u,u−v)≤M˜u−uu−v.hεhhhhhεh1ThusM˜u−uhε≤infu−vh1.α˜vh∈VhHeretheconstantM/˜α˜doesnotdependonε,h,andu.Thisestimateisweakerthanthestandardestimate,becausetheε-weightedH1-normisweakerthantheH1-norm.Moreover,theerrorofthebestapproximationis 9.2.TheStreamline-DiffusionMethod375notindependentofε,ingeneral.Forexample,ifweapplycontinuous,piece-wiselinearelements,then,undertheadditionalassumptionu∈H2(Ω),Theorem3.29yieldstheestimateinfu−vh1≤u−Ih(u)1≤Ch|u|2,vh∈VhwheretheconstantC>0doesnotdependonε,h,andu.So,wefinallyarriveattherelationu−uhε≤Ch|u|2.(9.13)However,theH2-seminormofthesolutionudependsonεinadisadvan-tageousmanner;forexample,itmaybe(cf.also[27,LemmaIII.1.18])that|u|=O(ε−3/2)(ε→0).2Thisresultissharp,sinceforexamplesofboundaryvalueproblemsforordinarylineardifferentialequationstheerrorofthebestapproximationalreadyexhibitsthisasymptoticbehaviour.Sothepracticalaswellasthetheoreticalproblemsmentionedaboveindicatethenecessitytousespecialnumericalmethodsforsolvingconvection-dominatedequations.Inthenextsections,asmallcollectionofthesemethodswillbedepicted.9.2TheStreamline-DiffusionMethodThestreamline-diffusionmethodistheprevalentmethodinthenumericaltreatmentofstationaryconvection-dominatedproblems.ThebasicideaisduetoBrooksandHughes[49],whocalledthemethodthestreamlineupwindPetrov–Galerkinmethod(SUPGmethod).Wedescribetheideaofthemethodforaspecialcaseofboundaryvalueproblem(9.2)underconsideration.LetthedomainΩ⊂Rdbeaboundedpolyhedron.Weconsiderthesamemodelasintheprecedingsection,thatis,K(x)≡εIwithaconstantcoefficientε>0,c∈C1(Ω,Rd),r∈C(Ω),f∈L2(Ω).Wealsoassumethattheinequalityr−1∇·c≥risvalidinΩ,20wherer0>0isaconstant.Thenthevariationalformulationof(9.2)readsasfollows:Findu∈Vsuchthata(u,v)=f,vforallv∈V,(9.14)0whereaisthebilinearform(9.9).Givenaregularfamilyoftriangulations{Th},letVh⊂Vdenotethesetofcontinuousfunctionsthatarepiecewisepolynomialofdegreek∈Nandsatisfytheboundaryconditions,i.e.,Vh:=vh∈Vvh|K∈Pk(K)forallK∈Th.(9.15) 3769.DiscretizationofConvection-DominatedProblemsIfinadditionthesolutionu∈Vof(9.14)belongstothespaceHk+1(Ω),wehave,by(3.87),thefollowingerrorestimatefortheinterpolantIh(u):k+1−lu−Ih(u)l,K≤cinthK|u|k+1,K(9.16)for0≤l≤k+1andallK∈Th.SincethespacesVhareoffinitedimension,aso-calledinverseinequalitycanbeproven(cf.Theorem3.43,(2)andExercise9.3):cinv∆vh0,K≤|vh|1,K(9.17)hKforallvh∈VhandallK∈Th.Hereitisimportantthattheconstantscint,cinv>0from(9.16)and(9.17),respectively,donotdependonuorvhandontheparticularelementsK∈Th.Thebasicideaofthestreamline-diffusionmethodconsistsintheadditionofsuitablyweightedresidualstothevariationalformulation(9.14).Becauseoftheassumptionu∈Hk+1(Ω),k∈N,thedifferentialequationcanbeinterpretedasanequationinL2(Ω).Inparticular,itisvalidonanyelementK∈TinthesenseofL2(K),i.e.,h−ε∆u+c·∇u+ru=falmosteverywhereinKandforallK∈Th.Nextwetakeanelementwisedefinedmappingτ:V→L2(Ω)andmultiplyhthelocaldifferentialequationinL2(K)bytherestrictionofτ(v)toK.hScalingbyaparameterδK∈RandsummingtheresultsoverallelementsK∈Th,weobtainδK−ε∆u+c·∇u+ru,τ(vh)0,K=δKf,τ(vh)0,K.K∈ThK∈ThIfweaddthisrelationtoequation(9.14)restrictedtoVh,weseethattheweaksolutionu∈V∩Hk+1(Ω)satisfiesthefollowingvariationalequation:ah(u,vh)=f,vhhforallvh∈Vh,whereah(u,vh):=a(u,vh)+δK−ε∆u+c·∇u+ru,τ(vh)0,K,K∈Thf,vhh:=f,v0+δKf,τ(vh)0,K.K∈ThThenthecorrespondingdiscretizationreadsasfollows:Finduh∈Vhsuchthatah(uh,vh)=f,vhhforallvh∈Vh.(9.18)Corollary9.2Supposetheproblems(9.14)and(9.18)haveasolutionu∈V∩Hk+1(Ω)andu∈V,respectively.Thenthefollowingerrorequationhhisvalid:ah(u−uh,vh)=0forallvh∈Vh.(9.19) 9.2.Streamline-DiffusionMethod377Inthestreamline-diffusionmethod(sdFEM),themappingτusedin(9.18)ischosenasτ(vh):=c·∇vh.Withoutgoingintodetails,wementionthatafurtheroptionistosetτ(vh):=−ε∆vh+c·∇vh+rvh.Thisresultsintheso-calledGalerkin/leastsquares–FEM(GLSFEM)[54].Especiallywithregardtotheextensionofthemethodtootherfiniteele-mentspaces,thediscussionofhowtochooseτandδKisnotyetcomplete.InterpretationoftheAdditionalTermintheCaseofLinearEle-mentsIfthefiniteelementspacesVhareformedbypiecewiselinearfunctions(i.e.,intheabovedefinition(9.15)ofVhwehavek=1),weget∆vh|K=0forallK∈Th.Ifinadditionthereisnoreactiveterm(i.e.,r=0),thediscretebilinearformis
ah(uh,vh)=ε∇uh·∇vhdx+c·∇uh,vh0+δKc·∇uh,c·∇vh0,K.ΩK∈ThSincethescalarproductappearinginthesumcanberewrittenasc·∇u,c·∇v=(ccT∇u)·∇vdx,weobtainthefollowinghh0,KKhhequivalentrepresentation:
a(u,v)=(εI+δccT)∇u·∇vdx+c·∇u,v.hhhKhhhh0KK∈ThThisshowsthattheadditionaltermintroducesanelement-dependentextradiffusioninthedirectionoftheconvectivefieldc(cf.alsoExercise0.3),whichmotivatesthenameofthemethod.Inthisrespect,thestreamline-diffusionmethodcanbeunderstoodasanimprovedversionofthefullupwindmethod,asseen,forexample,in(9.6).AnalysisoftheStreamline-DiffusionMethodTostarttheanalysisofstabilityandconvergencepropertiesofthestreamline-diffusionmethod,weconsiderthetermah(vh,vh)forarbitraryvh∈Vh.AsinSection3.2.1,thestructureofthediscretebilinearformahallowsustoderivetheestimatea(v,v)≥ε|v|2+rv2+δ−ε∆v+c·∇v+rv,c·∇v.hhhh10h0Khhhh0,KK∈ThFurthermore,neglectingforamomentthesecondterminthesumandusingtheelementaryinequalityab≤a2+b2/4forarbitrarya,b∈R,we 3789.DiscretizationofConvection-DominatedProblemsgetδK−ε∆vh+rvh,c·∇vh0,KK∈Th;<""≤−ε|δK|∆vh,|δK|c·∇vh0,KK∈Th;"<"+|δK|rvh,|δK|c·∇vh0,K≤ε2|δ|∆v2+|δ|r2v2Kh0,KK0,∞,Kh0,KK∈Th|δK|2+c·∇vh0,K.2Bymeansoftheinverseinequality(9.17)itfollowsthatc22inv2δK−ε∆vh+rvh,c·∇vh0,K≤ε|δK|2|vh|1,KhK∈ThK∈ThK22|δK|2+|δK|r0,∞,Kvh0,K+c·∇vh0,K.2Puttingthingstogether,weobtainc2a(v,v)≥ε−ε2|δ|inv|v|2v2hhhKh2h1,Kh0,KK∈ThK2|δK|2+r0−|δK|r0,∞,K+δK−c·∇vh0,K.2Thechoice$%1h2rK00<δK≤min2,2(9.20)2εcinvr0,∞,Kleadstoε2r0212ah(vh,vh)≥|vh|1+vh0+δKc·∇vh0,K.222K∈ThTherefore,iftheso-calledstreamline-diffusionnormisdefinedby$%1/2v:=ε|v|2+rv2+δc·∇v2,v∈V,sd100K0,KK∈Ththenthechoice(9.20)impliestheestimate12vhsd≤ah(vh,vh)forallvh∈Vh.(9.21)2 9.2.Streamline-DiffusionMethod379Obviously,thestreamline-diffusionnorm·sdisstrongerthantheε-weightedH1-norm(9.10);i.e.,√min{1,r0}vε≤vsdforallv∈V.Nowanerrorestimatecanbeproven.Sinceestimate(9.21)holdsonlyonthefiniteelementspacesVh,weconsiderfirstthenormofIh(u)−uh∈Vhandmakeuseoftheerrorequation(9.19):12Ih(u)−uhsd≤ah(Ih(u)−uh,Ih(u)−uh)=ah(Ih(u)−u,Ih(u)−uh).2Inparticular,undertheassumptionu∈V∩Hk+1(Ω)thefollowingthreeestimatesarevalid:
√ε∇(Ih(u)−u)·∇(Ih(u)−uh)dx≤ε|Ih(u)−u|1Ih(u)−uhsdΩ√≤cεhk|u|I(u)−u,intk+1hhsd
[c·∇(Ih(u)−u)+r(Ih(u)−u)](Ih(u)−uh)dxΩ
=(r−∇·c)(Ih(u)−u)(Ih(u)−uh)dxΩ
−(Ih(u)−u)c·∇(Ih(u)−uh)dxΩ≤r−∇·c0,∞Ih(u)−u0Ih(u)−uh0+Ih(u)−u0c·∇(Ih(u)−uh)0$%1/2≤CI(u)−u2h0,KK∈Th$%1/2−12+δKIh(u)−u0,KIh(u)−uhsdK∈Th$%1/2≤Chk1+δ−1h2|u|2I(u)−u,KKk+1,KhhsdK∈ThandδK−ε∆(Ih(u)−u)+c·∇(Ih(u)−u)K∈Th+r(Ih(u)−u),c·∇(Ih(u)−uh)0,K(9.22) 3809.DiscretizationofConvection-DominatedProblems"&'≤cδεhk−1+chk+rhk+1intKK0,∞,KK0,∞,KKK∈Th"×|u|k+1,KδKc·∇(Ih(u)−uh)0,K$%1/2&'2≤Cδεhk−1+hk+hk+1|u|2I(u)−u.KKKKk+1,KhhsdK∈ThCondition(9.20),whichwasalreadyrequiredforestimate(9.21),impliesthath2εδ≤K,K2cinvandsotheapplicationtothefirsttermofthelastboundleadsto3δK−ε∆(Ih(u)−u)+c·∇(Ih(u)−u)K∈Th4+r(Ih(u)−u),c·∇(Ih(u)−uh)0,K$%1/2≤Chk[ε+δ]|u|2I(u)−u.Kk+1,KhhsdK∈ThCollectingtheestimatesanddividingbyIh(u)−uhsd,weobtaintherelation$56%1/2h2I(u)−u≤Chkε+K+h2+δ|u|2.hhsdKKk+1,KδKK∈ThFinally,thetermsinthesquarebracketswillbeequilibratedwiththehelpofcondition(9.20).Werewritetheε-dependentterminthisconditionash22K=Pehεc2c2cKKinvinv∞,Kwithc∞,KhKPeK:=.(9.23)2εThislocalP´ecletnumberisarefinementofthedefinition(9.4).ThefollowingdistinctionsconcerningPeKareconvenient:PeK≤1andPeK>1.Inthefirstcase,wechooseh22δ=δPeh=δK,δ=δ,K0KK101εc∞,K 9.2.Streamline-DiffusionMethod381withappropriateconstantsδ0>0andδ1>0,respectively,whichareindependentofKandε.Thenwehaveh212Peε+K+h2+δ=1+ε+h2+δKh≤C(ε+h),KKK1KKδKδ1c0,∞,KwhereC>0isindependentofKandε.Inthesecondcase,itissufficienttochooseδK=δ2hKwithanappropriateconstantδ2>0thatisindependentofKandε.Thenδδch2220,∞,KKδK=PeKhK=PeK2PeKεandh21ε+K+h2+δ=ε++δh+h2≤C(ε+h),KK2KKKδKδ2withC>0independentofKandε.Notethatinbothcasestheconstantscanbechosensufficientlysmall,independentofPeK,thatthecondi-tion(9.20)issatisfied.Nowwearepreparedtoprovethefollowingerrorestimate.Theorem9.3LettheparametersδKbegivenbyh2δK,Pe≤1,δ=1KKεδ2hK,PeK>1,whereδ1,δ2>0donotdependonKandεandarechosensuchthatcondition(9.20)issatisfied.Iftheweaksolutionuof(9.14)belongstoHk+1(Ω),then√√u−u≤Cε+hhk|u|,hsdk+1wheretheconstantC>0isindependentofε,h,andu.Proof:Bythetriangleinequality,wegetu−uhsd≤u−Ih(u)sd+Ih(u)−uhsd.Anestimateofthesecondaddendisalreadyknown.Todealwiththefirstterm,theestimatesoftheinterpolationerror(9.16)areuseddirectly:2u−Ih(u)sd=ε|u−I(u)|2+ru−I(u)2+δc·∇(u−I(u))2h10h0Kh0,KK∈ThAB≤c2εh2k+rh2(k+1)+δc2h2k|u|2intK0KK0,∞,KKk+1,KK∈Th 3829.DiscretizationofConvection-DominatedProblems&'≤Ch2kε+h2+δ|u|2≤C(ε+h)h2k|u|2.KKKk+1,KKk+1K∈Th
Remark9.4(i)InthecaseoflargelocalP´ecletnumbers,wehaveε≤1chandthus2∞,KK$%1/22k+1/2u−uh0+δ2hKc·∇(u−uh)0,K≤Ch|u|k+1.K∈ThSotheL2-errorofthesolutionisnotoptimalincomparisonwiththeestimateoftheinterpolationerroru−I(u)≤Chk+1|u|,h0k+1whereastheL2-errorofthedirectionalderivativeofuinthedirectionofcisoptimal.(ii)Ingeneral,theseminorm|u|k+1dependsonnegativepowersofε.Therefore,ifh→0,theconvergenceinTheorem9.3isnotuniformwithrespecttoε.ComparingtheestimatefromTheorem9.3forthespecialcaseofcontinuouslinearelementswiththeestimate(9.13)forthecorrespondingstandardmethodgivenattheendoftheintroduction,i.e.,u−uhε≤Ch|u|2,weseethattheerrorofthestreamline-diffusionmethodismeasuredinastrongernormthanthe·ε-normandadditionally,thattheerrorboundisasymptoticallybetterintheinterestingcaseε0,rewritetheordinaryboundaryvalueproblem(−εu
+u)
=0inΩ:=(0,1),u(0)=u(1)−1=0,intoanequivalentformbutwithnonnegativeright-handsideandhomogeneousDirichletboundaryconditions.(b)ComputetheH2(0,1)-seminormofthesolutionofthetransformedproblemandinvestigateitsdependenceonε.9.2Provetheerrorequationofthestreamline-diffusionmethod(Corollary9.2).9.3Givenanarbitrary,butfixed,triangleKwithdiameterhK,provetheinequalitycinv∆p0,K≤|p|1,KhKforarbitrarypolynomialsp∈Pk(K),k∈N,wheretheconstantcinv>0isindependentofKandp.9.4Verifythatthestreamline-diffusionnorm·sdisindeedanorm.9.3FiniteVolumeMethodsIntheconvection-dominatedsituation,thefinitevolumemethodintro-ducedinChapter6provestobeaverystable,butnotsoaccurate,method.OnereasonforthisstabilityliesinanappropriateasymptoticbehaviouroftheweightingfunctionRforlargeabsolutevaluesofitsargument.Namely,ifweconsidertheexamplesofnonconstantweightingfunctionsgiveninSection6.2.2,weseethat(P4)limR(z)=0,limR(z)=1.z→−∞z→∞Inthegeneralcaseofthemodelproblem(6.5)withk=ε>0,(P4)γijdijimpliesthatfor−1thetermrijui+(1−rij)ujinthebilinearεγijdijformbheffectivelyequalsuj,whereasinthecase1thequantityεuiremains.Inotherwords,inthecaseofdominatingconvection,theapproximationbhevaluatesthe“information”(ujorui)upwind,i.e.,justatthatnode(ajorai)fromwhich“theflowiscoming”. 3849.DiscretizationofConvection-DominatedProblemsThisessentiallycontributestothestabilizationofthemethodandmakesitpossibletoprovepropertiessuchasglobalconservativityorinversemono-tonicity(cf.Section6.2.4)withoutanyrestrictionsonthesizeofthelocalγijdijP´ecletnumberandthuswithoutanyrestrictionsontheratioofhεandε.ThislocalP´ecletnumber(notethemissingfactor2incomparisonto(9.23))alsotakesthedirectionoftheflowcomparedtotheedgeaiajintoaccount.TheChoiceofWeightingParametersInordertomotivatethechoiceoftheweightingparametersinthecaseoftheVoronoidiagram,werecalltheessentialstepinthederivationofthefinitevolumemethod,namelytheapproximationoftheintegral
Iij:=[µij(νij·∇u)−γiju]dσ.ΓijItfirstsuggestsitselftoapplyasimplequadraturerule,forexampleIij≈qijmij,whereqijdenotesthevalueoftheexpressiontobeintegratedatthepointaijoftheintersectionoftheboundarysegmentΓijwiththeedgeboundedbytheverticesaiandaj(i.e.,2aij=ai+aj).Next,ifthisedgeisparametrisedaccordingto5611x=x(τ)=aij+τdijνij,τ∈−,,22andifweintroducethecompositefunctionw(τ):=u(x(τ)),thenwecanwriteµijdwµij(νij·∇u)−γiju=q(0)withq(τ):=(τ)−γijw(τ).dijdτTherelationdefiningthefunctionqcanbeinterpretedasalinearordinary&'differentialequationfortheunknownfunctionw:−1,1→R.Provided&'22thatqiscontinuousontheinterval−1,1,theequationcanbesolved22exactly:$
τ%dijγijdij11w(τ)=q(s)exp−s+ds+w−µij−1/2µij22γijdij1×expτ+.µij2Approximatingqbyaconstantqij,wegetinthecaseγij=0,56qijγijdij11w(τ)≈1−exp−τ++w−γijµij22γijdij1×expτ+.µij2 9.3.FiniteVolumeMethods385Inparticular,561qijγijdij1γijdijw≈1−exp−+w−exp;(9.24)2γijµij2µijthatis,theapproximationqijofq(0)canbeexpressedbymeansofthevaluesw(±1):2w(1)−w(−1)expγijdij22µijqij≈γij.(9.25)γijdijexp−1µijInthecaseγij=0,itimmediatelyfollowsfromtheexactsolutionandtheapproximationq≈qijthatw(1)−w(−1)q≈µ22.ijijdijSincethisisequaltothelimitof(9.25)forγij→0,wecanexclusivelyworkwiththerepresentation(9.25).IfwedefinetheweightingfunctionR:R→[0,1]by1zR(z):=1−1−,(9.26)zez−1γijdijthenwiththechoicerij:=Rµij,(9.25)canbewrittenasuj−uiqij≈µij−[rijui+(1−rij)uj]γij.dijAsimplealgebraicmanipulationshowsthatthisisexactlytheapproxima-tionschemegiveninSection6.2.Theuseoftheweightingfunction(9.26)yieldsadiscretizationmethodthatcanbeinterpretedasageneralizationoftheso-calledIl’in–Allen–Southwellscheme.However,inordertoavoidthecomparativelyexpensivecomputationofthefunctionvaluesrijof(9.26),oftensimplerfunctionsR:R→[0,1]areused(seeSection6.2.2),whicharetosomeextentapproximationsof(9.26)keepingtheproperties(P1)to(P4).Attheendofthisparagraphwewillillustratetheimportanceoftheprop-erties(P1)to(P3),especiallyforconvection-dominatedproblems.Property(P2)hasbeenusedintheproofofthebasicstabilityestimate(6.20).Ontheotherhand,wehaveseenatseveralplaces(e.g.,inSection1.4orinChapter5)thatthematrixAhofthecorrespondingsystemoflinearal-gebraicequationsshouldhavepositivediagonalentries.Forexample,ifinthedifferentialequationfrom(9.2)thereactiontermdisappears,thenproperties(P1)and(P3)guaranteethatthediagonalentriesareatleastnonnegative.Thiscanbeseenasfollows: 3869.DiscretizationofConvection-DominatedProblemsFrom(6.9)weconcludethefollowingformula:5656µijµijγijdij(Ah)ii=+γijrijmij=1+rijmij,i∈Λ.dijdijµijIfwereplaceinproperty(P3)thenumberzby−z,thenweget,byproperty(P1),0≤1+[1−R(−z)]z=1+zR(z).Therefore,iftheweightingfunctionRsatisfies(P1)and(P3),thenwehavethat(Ah)ii≥0foralli∈Λ.Thesimplechoicer≡1doesnotsatisfyproperty(P3).Inthiscase,ij2thecondition(Ah)ii≥0leadstotherequirementγijdij−≤1,2µijwhichinthecaseγij≤0,i.e.,foralocalflowfromajtoai,isarestrictiontotheratioofhandε,andthisisanalogoustothecondition(9.5)onthegridP´ecletnumber,whereonlythesizesofK,c,andhenter.Similarly,itcanbeshownthatproperty(P3)impliesthenonpositivityoftheoff-diagonalentriesofAh.AnErrorEstimateAttheendofthissectionanerrorestimatewillbecited,whichcanbederivedsimilarlytothecorrespondingestimateofthestandardmethod.Theonlyspecialaspectisthatthedependenceoftheoccurringquantitiesonεiscarefullytracked(see[40]).Theorem9.5Let{Th}hbearegularfamilyofconformingtriangula-tions,alltrianglesofwhicharenonobtuse.Furthermore,inadditiontotheassumptionsonthecoefficientsofthebilinearform(9.9),letf∈C1(Ω).IftheexactsolutionuofthemodelproblembelongstoH2(Ω)andifuh∈Vhdenotestheapproximativesolutionofthefinitevolumemethod(6.11),wheretheapproximationsγij,respectivelyri,arechosenaccordingto(6.7),respectively(6.8),thenforsufficientlysmallh>¯0theestimatehu−uhε≤C√[u2+|f|1,∞],h∈(0,¯h],εholds,whereboththeconstantC>0andh>¯0donotdependonε.Inspecial,butpracticallynotsorelevant,cases(forexample,ifthetrian-gulationsareofFriedrichs–Kellertype),itispossibletoremovethefactor√1intheboundabove.εComparingthefinitevolumemethodwiththestreamline-diffusionmethod,weseethatthefinitevolumemethodislessaccurate.However,itisgloballyconservativeandinversemonotone. 9.4.TheLagrange–GalerkinMethod387Exercise9.5Usinganequidistantgrid,formulateboththestreamline-diffusionmethodandthefinitevolumemethodforaone-dimensionalmodelproblem(d=1,Ω=(0,1),r=0)withconstantcoefficientsandcomparethere-sultingdiscretizations.Basedonthatcomparison,whatcanbesaidaboutthechoiceoftheparametersinthestreamline-diffusionmethod?9.4TheLagrange–GalerkinMethodIntheprevioussections,discretizationmethodsforstationarydiffusion-convectionequationswerepresented.Inconjunctionwiththemethodoflines,thesemethodscanalsobeappliedtoparabolicproblems.However,sincethemethodoflinesdecouplesspatialandtemporalvariables,itcan-notbeexpectedthatthepeculiaritiesofnonstationarydiffusion-convectionequationsarereflectedadequately.Theso-calledLagrange–Galerkinmethodattemptstobypassthisprob-lembymeansofanintermediatechangefromtheEuleriancoordinates(considereduptonow)totheso-calledLagrangiancoordinates.Thelatterarechoseninsuchawaythattheoriginofthecoordinatesystem(i.e.,thepositionoftheobserver)ismovedwiththeconvectivefield,andinthenewcoordinatesnoconvectionoccurs.Toillustratethebasicidea,thefollowinginitial-boundaryvalueproblemwillbeconsidered,whereΩ⊂RdisaboundeddomainwithLipschitzcontinuousboundaryandT>0:Forgivenfunctionsf:QT→Randu0:Ω→R,findafunctionu:QT→Rsuchthat∂u+Lu=finQT,∂tu=0onST,(9.27)u=u0onΩ×{0},where(Lu)(x,t):=−∇·(K(x)∇u(x,t))+c(x,t)·∇u(x,t)+r(x,t)u(x,t),(9.28)withsufficientlysmoothcoefficientsK:Ω→Rd,d,c:Q→Rd,r:Q→R.TTAsusual,thedifferentialoperators∇and∇·actonlywithrespecttothespatialvariables.Thenewcoordinatesystemisobtainedbysolvingthefollowingparameter-dependentauxiliaryproblem: 3889.DiscretizationofConvection-DominatedProblemsGiven(x,s)∈Q,findavectorfieldX:Ω×[0,T]2→RdsuchthatTdX(x,s,t)=c(X(x,s,t),t),t∈(0,T),dt(9.29)X(x,s,s)=x.ThetrajectoriesX(x,s,·)arecalledcharacteristics(through(x,s)).IfciscontinuousonQTand,forfixedt∈[0,T],LipschitzcontinuouswithrespecttothefirstargumentonΩ,thenthereexistsauniquesolutionX=X(x,s,t).Denotingbyuthesufficientlysmoothsolutionof(9.27)andsettinguˆ(x,t):=u(X(x,s,t),t)forfixeds∈[0,T],thenthechainruleimpliesthat∂uˆ∂u(x,t)=+c·∇u(X(x,s,t),t).∂t∂tTheparticularvalue∂uˆ∂u(x,s)=(x,s)+c(x,s)·∇u(x,s)∂t∂tiscalledthematerialderivativeofuat(x,s).Thusthedifferentialequationreadsas∂uˆ−∇·(K∇u)+ru=f;∂ti.e.,itisformallyfreeofanyconvectiveterms.Nowtheequationwillbesemidiscretizedbymeansofthehorizontalmethodoflines.Atypicalwayistoapproximatethetimederivativebybackwarddifferencequotients.Soletanequidistantpartitionofthetimeinterval(0,T)withstepsizeτ:=T/N,N∈N(providedthatT<∞),begiven.Trackingthecharacteristicsbackwardsintime,inthestripΩ×[tn,tn+1),n∈{0,1,...,N−1},withx=X(x,tn+1,tn+1)thefollowingapproximationresults:∂uˆ11≈[ˆu(x,tn+1)−uˆ(x,tn)]=[u(x,tn+1)−u(X(x,tn+1,tn),tn)].∂tττFurther,ifVhdenotesafinite-dimensionalsubspaceofVinwhichwewanttofindtheapproximationstou(·,tn),themethodreadsasfollows:Givenu∈V,findanelementUn+1∈V,n∈{0,...,N−1},such0hhhthat134Un+1−Un(X(·,t,t)),vτn+1nh03434+K∇Un+1·∇v,1+r(·,t)Un+1,v=f(·,t),vh0n+1h0n+1h0forallvh∈Vh,U0=u.0h(9.30) 9.4.Lagrange–GalerkinMethod389Apossibleextensionofthemethodistousetime-dependentsubspaces;thatis,givenasequenceofsubspacesVn⊂V,n∈{0,...,N},thehapproximationsUntou(·,t)arechosenfromVn.nhSothebasicideaoftheLagrange–Galerkinmethod,namely,theelimi-nationofconvectivetermsbymeansofanappropriatetransformationofcoordinates,allowstheapplicationofstandarddiscretizationmethodsandmakesthemethodattractiveforsituationswhereconvectionisdominating.Infact,thereexistsawholevarietyofpapersdealingwitherroresti-matesforthemethodintheconvection-dominatedcase,butoftenundertheconditionthatthesystem(9.29)isintegratedexactly.Inpractice,theexactintegrationisimpossible,andthesystem(9.29)hastobesolvednumerically(cf.[61]).Thismayleadtostabilityproblems,sothereisstillaconsiderableneedinthetheoreticalfoundationofLagrange–Galerkinmethods.OnlyrecentlyithasbeenpossibleforamodelsituationtoproveorderofconvergenceestimatesuniformlyintheP´ecletnumberfor(9.30)(see[43]).ThekeyistheconsequentuseofLagrangiancoordinates,reveal-ingthat(9.30)isjusttheapplicationoftheimplicitEulermethodtoanequationarisingfromatransformationbycharacteristicsdefinedpiecewisebackwardintime.Thisequationisapurediffusionproblem,butwithacoefficientreflectingthetransformation.InconjunctionwiththebackwardEulermethodthisisnotvisibleintheellipticparttobediscretized.Thustrackingthecharacteristicsbackwardintimeturnsouttobeimportant. AAppendicesA.1NotationCsetofcomplexnumbersNsetofnaturalnumbersN0:=N∪{0}QsetofrationalnumbersRsetofrealnumbersR+setofpositiverealnumbersZsetofintegerszrealpartofthecomplexnumberzzimaginarypartofthecomplexnumberzxTtransposeofthevectorx∈Rd,d∈N1/p|x|:=d|x|p,x=(x,...,x)T∈Rd,d∈N,p∈[1,∞)pj=1j1d|x|:=max|x|maximumnormofthevectorx∈Rd,d∈N∞j=1,...,dj|x|:=|x|Euclideannormofthevectorx∈Rd,d∈N2Tddx·y:=xy=j=1xjyjscalarproductofthevectorsx,y∈Rx,y:=yTAx=y·Axenergyproductofthevectorsx,y∈Rdw.r.t.Aasymmetric,positivedefinitematrixA|α|:=|α|order(orlength)ofthemulti-indexα∈Nd,d∈N10IidentitymatrixoridentityoperatorejthunitvectorinRm,j=1,...,mjdiag(λ)=diag(λ,...,λ)diagonalmatrixinRm,mwithdiagonali1mentriesλ1,...,λm∈C A.1.Notation391ATtransposeofthematrixAA−TtransposeoftheinversematrixA−1detAdeterminantofthesquarematrixAλmin(A)minimumeigenvalueofamatrixAwithrealeigenvaluesλmax(A)maximumeigenvalueofamatrixAwithrealeigenvaluesσ(A)setofeigenvalues(spectrum)ofthesquarematrixA(A)spectralradiusofthesquarematrixAm(A)bandwidthofthesymmetricmatrixAEnv(A)hullofthesquarematrixAp(A)profileofthesquarematrixAB(x0):={x:x−x0<}openballinanormedspaceB(x0):={x:x−x0≤}closedballinanormedspacediam(G)diameterofthesetG⊂Rd|G|n-dimensional(Lebesgue)measureoftheG⊂Rn,n∈{1,...,d}n|G|:=|G|d-dimensional(Lebesgue)measureofthesetG⊂Rddvol(G)length(d=1),area(d=2),volume(d=3)of“geometricbodies”G⊂RdintGinteriorofthesetG∂GboundaryofthesetGGclosureofthesetGspanGlinearhullofthesetGconvGconvexhullofthesetG|G|cardinalnumberofthediscretesetGνouterunitnormalw.r.t.thesetG⊂RdΩdomainofRd,d∈NΓ:=∂ΩboundaryofthedomainΩ⊂Rdsuppϕsupportofthefunctionϕf−1inverseofthemappingff[G]imageofthesetGunderthemappingff−1[G]preimageofthesetGunderthemappingff|Krestrictionoff:G→RtoasubsetK⊂GvXnormoftheelementvofthenormedspaceXdimXdimensionofthefinite-dimensionallinearspaceXL[X,Y]setoflinear,continuousoperatorsactingfromthenormedspaceXinthenormedspaceYX
:=L[X,R]dualspaceoftherealnormedspaceXO(·),o(·)Landausymbolsofasymptoticanalysisδij(i,j∈N0)Kroneckersymbol,i.e.,δii=1andδij=0ifi=jDifferentialexpressions∂l(l∈N)symbolforthepartialderivativew.r.t.thelthvariable∂t(t∈R)symbolforthepartialderivativew.r.t.thevariablet∂α(α∈Ndmulti-index)αthpartialderivative0∇:=(∂,...,∂)TNablaoperator(symbolicvector)1d 392A.Appendices∆Laplaceoperator∂µ:=µ·∇directionalderivativew.r.t.thevectorµDΦ:=∂Φ:=(∂Φ)mJacobimatrixorfunctionalmatrix∂xjii,j=1ofadifferentiablemappingΦ:Rm→RmCoefficientsindifferentialexpressionsKdiffusioncoefficient(asquarematrixfunction)cconvectioncoefficient(avectorfunction)rreactioncoefficientDiscretizationmethodsVhansatzspaceXhextendedansatzspacewithoutanyhomogeneousDirichletboundaryconditionsahapproximatedbilinearformbhapproximatedlinearformFunctionspaces(seealsoAppendixA.5)P(G)setofpolynomialsofmaximumdegreekonG⊂RdkC(G)=C0(G)setofcontinuousfunctionsonGCl(G)(l∈N)setofl-timescontinuouslydifferentiablefunctionsonGC∞(G)setofinfinitelyoftencontinuouslydifferentiablefunctionsonGC(G)=C0(G)setofboundedanduniformlycontinuousfunctionsonGCl(G)(l∈N)setoffunctionswithboundedanduniformlycontinuousderivativesuptotheorderlonGC∞(G)setoffunctions,allpartialderivativesofwhichareboundedanduniformlycontinuousonGC(G)=C0(G)setofcontinuousfunctionsonGwithcompactsupport00Cl(G)(l∈N)setofl-timescontinuouslydifferentiablefunctionsonG0withcompactsupportC∞(G)setofinfinitelyoftencontinuouslydifferentiablefunctionsonG0withcompactsupportLp(G)(p∈[1,∞))setofLebesgue-measurablefunctionswhosepthpoweroftheirabsolutevalueisLebesgue-integrableonGL∞(G)setofmeasurable,essentiallyboundedfunctions·,·scalarproductinL2(G)†0,G·norminL2(G)†0,G·(p∈[1,∞])norminLp(G)†0,p,G·norminL∞(G)†∞,GWl(G)(l∈N,p∈[1,∞])setofl-timesweaklydifferentiablefunctionspfromL(G),withderivativesinLp(G)p·(l∈N,p∈[1,∞])norminWl(G)†l,p,Gp|·|(l∈N,p∈[1,∞])seminorminWl(G)†l,p,Gp A.2.BasicConceptsofAnalysis393Hl(G):=Wl(G)(l∈N)2·,·(l∈N)scalarproductinHl(G)†l,G·(l∈N)norminHl(G)†l,G|·|(l∈N)seminorminHl(G)†l,G·,·discreteL2(Ω)-scalarproduct0,h·discreteL2(Ω)-norm0,hL2(∂G)setofsquareLebesgue-integrablefunctionsontheboundary∂GH1(G)setoffunctionsfromH1(G)withvanishingtraceon∂G0C([0,T],X)=C0([0,T],X)setofcontinuousfunctionson[0,T]withvaluesinthenormedspaceXCl([0,T],X)(l∈N)setofl-timescontinuouslydifferentiablefunctionson[0,T]withvaluesinthenormedspaceXLp((0,T),X)(p∈[1,∞])Lebesgue-spaceoffunctionson[0,T]withvaluesinthenormedspaceX†Convention:InthecaseG=Ω,thisspecificationisomitted.A.2BasicConceptsofAnalysisAsubsetG⊂Rdiscalledasetofmeasurezeroif,foranynumberε>0,acountablefamilyofballsBjwithd-dimensionalvolumeεj>0existssuchthat∞#∞εj<εandG⊂Bj.j=1j=1Twofunctionsf,g:G→Rarecalledequalalmosteverywhere(inshort:equala.e.,notation:f≡g)iftheset{x∈G:f(x)=g(x)}isofmeasurezero.Inparticular,afunctionf:G→Riscalledvanishingalmosteverywhereifitisequaltotheconstantfunctionzeroalmosteverywhere.Afunctionf:G→Riscalledmeasurableifthereexistsasequence(fi)iofstepfunctionsfi:G→Rsuchthatfi→ffori→∞almosteverywhere.Inwhatfollows,GdenotesasubsetofRd,d∈N.(i)Apointx=(x,x,...,x)T∈RdiscalledaboundarypointofG12difeveryopenneighbourhood(perhapsanopenball)ofxcontainsapointofGaswellasapointofthecomplementarysetRG.(ii)ThecollectionofallboundarypointsofGiscalledtheboundaryofGandisdenotedby∂G.(iii)ThesetG:=G∪∂GiscalledtheclosureofG.(iv)ThesetGiscalledclosedifG=G. 394A.Appendices(v)ThesetGiscalledopenifG∩∂G=∅.(vi)ThesetG∂GiscalledtheinteriorofGandisdenotedbyintG.AsubsetG⊂Rdiscalledconnectedifforarbitrarydistinctpointsx1,x2∈GthereexistsacontinuouscurveinGconnectingthem.ThesetGiscalledconvexifanytwopointsfromGcanbeconnectedbyastraight-linesegmentinG.Anonempty,open,andconnectedsetG⊂RdiscalledadomaininRd.Byα=(α,...,α)T∈Ndaso-calledmulti-indexisdenoted.Multi-1d0indicesareapopulartooltoabbreviatesomeelaboratenotation.Forexample,/d/dd∂α:=∂αi,α!:=α!,|α|:=α.iiii=1i=1i=1Thenumber|α|iscalledtheorder(orlength)ofthemulti-indexα.Foracontinuousfunctionϕ:G→R,thesetsuppϕ:={x∈G:ϕ(x)=0}denotesthesupportofϕ.A.3BasicConceptsofLinearAlgebraAsquarematrixA∈Rn,nwithentriesaiscalledsymmetricifa=aijijjiholdsforalli,j∈{1,...,n}.AmatrixA∈Rn,niscalledpositivedefiniteifx·Ax>0forallx∈Rn{0}.Givenapolynomialp∈Pk,k∈N0,oftheformkp(z)=azjwitha∈C,j∈{0,...,k}jjj=0andamatrixA∈Cn,n,thenthefollowingmatrixpolynomialofAcanbeestablished:kp(A):=aAj.jj=0EigenvaluesandEigenvectorsLetA∈Cn,n.Anumberλ∈CiscalledaneigenvalueofAifdet(A−λI)=0.IfλisaneigenvalueofA,thenanyvectorx∈Cn{0}suchthatAx=λx(⇔(A−λI)x=0)iscalledaneigenvectorofAassociatedwiththeeigenvalueλ. A.3.LinearAlgebra395ThepolynomialpA(λ):=det(A−λI)iscalledthecharacteristicpolynomialofA.ThesetofalleigenvaluesofamatrixAiscalledthespectrumofA,denotedbyσ(A).IfalleigenvaluesofamatrixAarereal,thenthenumbersλmax(A)andλmin(A)denotethelargest,respectivelysmallest,oftheseeigenvalues.Thenumber(A)=maxλ∈σ(A)|λ|iscalledthespectralradiusofA.NormsofVectorsandMatricesThenormofavectorx∈Rn,n∈N,isareal-valuedfunctionx→|x|satisfyingthefollowingthreeproperties:(i)|x|≥0forallx∈Rn,|x|=0⇔x=0,(ii)|αx|=|α||x|forallα∈R,x∈Rn,(iii)|x+y|≤|x|+|y|forallx,y∈Rn.Forexample,themostfrequentlyusedvectornormsare(a)themaximumnorm:|x|∞:=max|xj|.(A3.1)j=1...n(b)thep-norm,p∈[1,∞):$%1/pn|x|:=|x|p.(A3.2)pjj=1Theimportantcasep=2yieldstheso-calledEuclideannorm:$%1/2n|x|:=x2.(A3.3)2jj=1Thethreemostimportantnorms(thatis,p=1,2,∞)inRnareequivalentinthefollowingsense:Theinequalities1√√|x|2≤|x|∞≤|x|2≤n|x|∞,n1|x|1≤|x|∞≤|x|1≤n|x|∞,n1√√|x|1≤|x|2≤|x|1≤n|x|2narevalidforallx∈Rn.ThenormofthematrixA∈Rn,nisareal-valuedfunctionA→Asatisfyingthefollowingfourproperties:(i)A≥0forallA∈Rn,n,A=0⇔A=0,(ii)αA=|α|Aforallα∈R,A∈Rn,n,(iii)A+B≤A+BforallA,B∈Rn,n,(iv)AB≤ABforallA,B∈Rn,n. 396A.AppendicesIncomparisonwiththedefinitionofavectornorm,weincludehereanadditionalproperty(iv),whichiscalledthesubmultiplicativeproperty.Itrestrictsthegeneralsetofmatrixnormstothepracticallyimportantclassofsubmultiplicativenorms.Themostcommonmatrixnormsare(a)thetotalnorm:AG:=nmax|aik|,(A3.4)1≤i,k≤n(b)theFrobeniusnorm:$n%1/22AF:=aik,(A3.5)i,k=1(c)themaximumrowsum:nA∞:=max|aik|,(A3.6)1≤i≤nk=1(d)themaximumcolumnsum:nA1:=max|aik|.(A3.7)1≤k≤ni=1Allthesematrixnormsareequivalent.Forexample,wehave1AG≤Ap≤AG≤nAp,p∈{1,∞},nor1AG≤AF≤AG≤nAF.nNotethatthespectralradius(A)isnotamatrixnorm,asthefollowingsimpleexampleshows:01ForA=,wehavethatA=0but(A)=0.00However,foranymatrixnorm·thefollowingrelationisvalid:(A)≤A.(A3.8)Veryoften,matricesandvectorssimultaneouslyappearasaproductAx.Inordertobeabletohandlesuchsituations,thereshouldbeacertaincorrelationbetweenmatrixandvectornorms.Amatrixnorm·iscalledmutuallyconsistentorcompatiblewiththevectornorm|·|iftheinequality|Ax|≤A|x|(A3.9)isvalidforallx∈RnandallA∈Rn,n. A.3.LinearAlgebra397ExamplesofmutuallyconsistentnormsareAGorA∞with|x|∞,AGorA1with|x|1,AGorAFwith|x|2.Inmanycases,theboundfor|Ax|givenby(A3.9)isnotsharpenough;i.e.,forx=0wejusthavethat|Ax|<A|x|.Therefore,thequestionarisesofhowtofind,foragivenvectornorm,acompatiblematrixnormsuchthatin(A3.9)theequalityholdsforatleastoneelementx=0.Givenavectornorm|x|,thenumber|Ax|A:=sup=sup|Ax|x∈Rn{0}|x|x∈Rn:|x|=1iscalledtheinducedorsubordinatematrixnorm.Theinducednormisacompatiblematrixnormwiththegivenvectornorm.Itisthesmallestnormamongallmatrixnormsthatarecompatiblewiththegivenvectornorm|x|.Toillustratethedefinitionoftheinducedmatrixnorm,thematrixnorminducedbytheEuclideanvectornormisderived:(((A2:=max|Ax|2=maxxT(ATA)x=λmax(ATA)=(ATA).|x|2=1|x|2=1(A3.10)ThematrixnormA2inducedbytheEuclideanvectornormisalsocalledthespectralnorm.ThistermbecomesunderstandableinthespecialcaseofasymmetricmatrixA.Ifλ1,...,λndenotetherealeigenvaluesofA,thenthematrixATA=A2hastheeigenvaluesλ2satisfyingiA2=|λmax(A)|.Forsymmetricmatrices,thespectralnormcoincideswiththespectralra-dius.Becauseof(A3.8),itisthesmallestpossiblematrixnorminthatcase.Asafurtherexample,themaximumrowsumA∞isthematrixnorminducedbythemaximumnorm|x|∞.Thenumberκ(A):=AA−1iscalledtheconditionnumberofthematrixAwithrespecttothematrixnormunderconsideration. 398A.AppendicesThefollowingrelationholds:−1−11≤I=AA≤AA.For|·|=|·|p,theconditionnumberisalsodenotedbyκp(A).IfalleigenvaluesofAarereal,thenumberκ(A):=λmax(A)/λmin(A)iscalledthespectralconditionnumber.Hence,forasymmetricmatrixAtheequalityκ(A)=κ2(A)isvalid.Occasionally,itisnecessarytoestimatesmallperturbationsofnonsin-gularmatrices.Forthispurpose,thefollowingresultisuseful(perturbationlemmaorNeumann’slemma).LetA∈Rn,nsatisfyA<1withrespecttoanarbitrary,butfixed,matrixnorm.ThentheinverseofI−Aexistsandcanberepresentedasaconvergentpowerseriesoftheform∞(I−A)−1=Aj,j=0with−11(I−A)≤.(A3.11)1−ASpecialMatricesThematrixA∈Rn,niscalledanupper,respectivelylower,triangularmatrixifitsentriessatisfyaij=0fori>j,respectivelyaij=0forij+1).ThematrixA∈Rn,nsatisfiesthestrictrowsumcriterion(orisstrictlyrowdiagonallydominant)ifitsatisfiesn|aij|<|aii|foralli=1,...,n.j=1j=iItsatisfiesthestrictcolumnsumcriterionifthefollowingrelationholds:n|aij|<|ajj|forallj=1,...,n.i=1i=j A.4.SomeDefinitionsandArgumentsofLinearFunctionalAnalysis399ThematrixA∈Rn,nsatisfiestheweakrowsumcriterion(orisweaklyrowdiagonallydominant)ifn|aij|≤|aii|holdsforalli=1,...,nj=1j=iandthestrictinequality“<”isvalidforatleastonenumberi∈{1,...,n}.Theweakcolumnsumcriterionisdefinedsimilarly.ThematrixA∈Rn,niscalledreducibleifthereexistsubsetsN,N⊂12{1,...,n}withN1∩N2=∅,N1=∅=N2,andN1∪N2={1,...,n}suchthatthefollowingpropertyissatisfied:Foralli∈N1,j∈N2:aij=0.Amatrixthatisnotreducibleiscalledirreducible.AmatrixA∈Rn,niscalledanL-matrixiffori,j∈{1,...,n}the0inequalitiesaii≥0andaij≤0(i=j)arevalid.AnL0-matrixiscalledanL-matrixifalldiagonalentriesarepositive.AmatrixA∈Rn,niscalledmonotone(orofmonotonetype)iftherelationAx≤Ayfortwo(otherwisearbitrary)elementsx,y∈Rnimpliesx≤y.Heretherelationsignistobeunderstoodcomponentwise.Amatrixofmonotonetypeisinvertible.AmatrixA∈Rn,nisamatrixofmonotonetypeifitisinvertibleandallentriesoftheinversearenonnegative.Animportantsubclassofmatricesofmonotonetypeisformedbytheso-calledM-matrices.AmonotonematrixAwithaij≤0fori=jiscalledanM-matrix.LetA∈Rn,nbeamatrixwitha≤0fori=janda≥0(i,j∈ijii{1,...,n}).Inaddition,letAsatisfyoneofthefollowingconditions:(i)Asatisfiesthestrictrowsumcriterion.(ii)Asatisfiestheweakrowsumcriterionandisirreducible.ThenAisanM-matrix.A.4SomeDefinitionsandArgumentsofLinearFunctionalAnalysisWorkingwithvectorspaceswhoseelementsare(classicalorgeneralized)functions,itisdesirabletohaveameasureforthe“length”or“magnitude”ofafunction,and,asaconsequence,forthedistanceoftwofunctions. 400A.AppendicesLetVbearealvectorspace(inshort,anRvectorspace)andlet·beareal-valuedmapping·:V→R.Thepair(V,·)iscalledanormedspace(“Visendowedwiththenorm·”)ifthefollowingpropertieshold:u≥0forallu∈V,u=0⇔u=0,(A4.1)αu=|α|uforallα∈R,u∈V,(A4.2)u+v≤u+vforallu,v∈V.(A4.3)Theproperty(A4.1)iscalleddefiniteness;(A4.3)iscalledthetriangleinequality.Ifamapping·:V→Rsatisfiesonly(A4.2)and(A4.3),itiscalledaseminorm.Dueto(A4.2),westillhave0=0,buttheremayexistelementsu=0withu=0.AparticularlyinterestingexampleofanormcanbeobtainedifthespaceVisequippedwithaso-calledscalarproduct.Thisisamapping·,·:V×V→Rwiththefollowingproperties:(1)·,·isabilinearform,thatis,u,v1+v2=u,v1+u,v2forallu,v1,v2∈V,(A4.4)u,αv=αu,vforallu,v∈V,α∈R,andananalogousrelationisvalidforthefirstargument.(2)·,·issymmetric,thatis,u,v=v,uforallu,v∈V.(A4.5)(3)·,·ispositive,thatis,u,u≥0forallu∈V.(A4.6)(4)·,·isdefinite,thatis,u,u=0⇔u=0.(A4.7)Apositiveanddefinitebilinearformiscalledpositivedefinite.Ascalarproduct·,·definesanormonVinanaturalwayifweset1/2v:=v,v.(A4.8)Inabsenceofthedefiniteness(A4.7),onlyaseminormisinduced.Anorm(oraseminorm)inducedbyascalarproduct(respectivelybyasymmetricandpositivebilinearform)hassomeinterestingproperties.Forexample,itsatisfiestheCauchy–Schwarzinequality,thatis,|u,v|≤uvforallu,v∈V,(A4.9)andtheparallelogramidentity2222u+v+u−v=2(u+v)forallu,v∈V.(A4.10) A.4.LinearFunctionalAnalysis401TypicalexamplesofnormedspacesarethespacesRnequippedwithoneofthep-norms(forsomefixedp∈[1,∞]).Inparticular,theEuclideannorm(A3.3)isinducedbytheEuclideanscalarproductn(x,y)→x·yforallx,y∈R.(A4.11)Ontheotherhand,infinite-dimensionalfunctionspacesplayanimportantrole(seeAppendixA.5).IfavectorspaceVisequippedwithascalarproduct·,·,then,inanalogytoRn,anelementu∈Vissaidtobeorthogonaltov∈Vifu,v=0.(A4.12)Givenanormedspace(V,·),itiseasytodefinetheconceptofconvergenceofasequence(ui)iinVtou∈V:ui→ufori→∞⇐⇒ui−u→0fori→∞.(A4.13)Often,itisnecessarytoconsiderfunctionspacesendowedwithdifferentnorms.Insuchsituations,differentkindsofconvergencemayoccur.How-ever,ifthecorrespondingnormsareequivalent,thenthereisnochangeinthetypeofconvergence.Twonorms·1and·2inVarecalledequivalentifthereexistconstantsC1,C2>0suchthatC1u1≤u2≤C2u1forallu∈V.(A4.14)Ifthereisonlyaone-sidedinequalityoftheformu2≤Cu1forallu∈V(A4.15)withaconstantC>0,thenthenorm·1iscalledstrongerthanthenorm·2.Inafinite-dimensionalvectorspace,allnormsareequivalent.ExamplescanbefoundinAppendixA.3.Inparticular,itisimportanttoobservethattheconstantsmaydependonthedimensionnofthefinite-dimensionalvectorspace.Thisobservationalsoindicatesthatinthecaseofinfinite-dimensionalvectorspaces,theequivalenceoftwodifferentnormscannotbeexpected,ingeneral.Asaconsequenceof(A4.14),twoequivalentnorms·1,·2inVyieldthesametypeofconvergence:ui→uw.r.t.·1⇔ui−u1→0⇔ui−u2→0⇔ui→uw.r.t.·2.(A4.16)Inthisbook,thefinite-dimensionalvectorspaceRnisusedintwoas-pects:Forn=d,itisthebasicspaceofindependentvariables,andforn=Morn=mitrepresentsthefinite-dimensionaltrialspace.Inthefirstcase,theequivalenceofallnormscanbeusedinallestimateswithoutanysideeffects,whereasinthesecondcasetheaimistoobtainuniform 402A.AppendicesestimateswithrespecttoallMandm,andsothedependenceoftheequivalenceconstantsonMandmhastobefollowedthoroughly.Nowweconsidertwonormedspaces(V,·V)and(W,·W).Amappingf:V→Wiscalledcontinuousinv∈Vifforallsequences(vi)iinVwithvi→vfori→∞wegetf(vi)→f(v)fori→∞.Notethatthefirstconvergenceismeasuredin·Vandthesecondonein·W.Henceachangeofthenormmayhaveaninfluenceonthecontinuity.Asinclassicalanalysis,wecansaythatfiscontinuousinallv∈V⇐⇒(A4.17)f−1[G]isclosedforeachclosedG⊂W.Here,asubsetG⊂WofanormedspaceWiscalledclosedifforanysequence(ui)ifromGsuchthatui→ufori→∞theinclusionu∈Gfollows.Becauseof(A4.17),theclosednessofasetcanbeverifiedbyshowingthatitisacontinuouspreimageofaclosedset.Theconceptofcontinuityisaqualitativerelationbetweenthepreimageandtheimage.AquantitativerelationisgivenbythestrongernotionofLipschitzcontinuity:Amappingf:V→WiscalledLipschitzcontinuousifthereexistsaconstantL>0,theLipschitzconstant,suchthatf(u)−f(v)W≤Lu−vVforallu,v∈V.(A4.18)slope:Ladmissibleregionforf(y)fslope:-LxFigureA.1.Lipschitzcontinuity(forV=W=R).ALipschitzcontinuousmappingwithL<1iscalledcontractiveoracontraction;cf.FigureA.1.Mostofthemappingsusedarelinear;thatis,theysatisfy%f(u+v)=f(u)+f(v),forallu,v∈Vandλ∈R.(A4.19)f(λu)=λf(u),Foralinearmapping,theLipschitzcontinuityisequivalenttotheboundedness;thatis,thereexistsaconstantC>0suchthatf(u)W≤CuVforallu∈V.(A4.20) A.4.LinearFunctionalAnalysis403Infact,foralinearmappingf,thecontinuityatonepointisequivalentto(A4.20).Linear,continuousmappingsactingfromVtoWarealsocalled(linear,continuous)operatorsandaredenotedbycapitalletters,forexampleS,T,....InthecaseV=W=Rn,thelinear,continuousoperatorsinRnarethemappingsx→AxdefinedbymatricesA∈Rn,n.Theirboundedness,forexamplewithrespectto·V=·W=·∞,isanimmediateconsequenceofthecompatibilitypropertyofthe·∞-norm.Moreover,sinceallnormsinRnareequivalent,thesemappingsareboundedwithrespecttoanynormsinRn.Similarlyto(A4.20),abilinearformf:V×V→Riscontinuousifitisbounded,thatis,ifthereexistsaconstantC>0suchthat|f(u,v)|≤CuVvVforallu,v∈V.(A4.21)Inparticular,dueto(A4.9)anyscalarproductiscontinuouswithrespecttotheinducednormofV;thatis,ui→u,vi→v⇒ui,vi→u,v.(A4.22)Nowlet(V,·V)beanormedspaceandWasubspacethatis(addi-tionallyto·V)endowedwiththenorm·W.Theembeddingfrom(W,·W)to(V,·V),i.e.,thelinearmappingthatassignsanyelementofWtoitselfbutconsideredasanelementofV,iscontinuousiffthenorm·Wisstrongerthanthenorm·V(cf.(A4.15)).Thecollectionoflinear,continuousoperatorsfrom(V,·V)to(W,·W)formsanRvectorspacewiththefollowing(argumentwise)operations:(T+S)(u):=T(u)+S(u)forallu∈V,(λT)(u):=λT(u)forallu∈V,foralloperatorsT,Sandλ∈R.ThisspaceisdenotedbyL[V,W].(A4.23)InthespecialcaseW=R,thecorrespondingoperatorsarecalledlinear,continuousfunctionals,andthenotation
V:=L[V,R](A4.24)isused.TheRvectorspaceL[V,W]canbeequippedwithanorm,theso-calledoperatornorm,byT:=supT(u)Wu∈V,uV≤1forT∈L[V,W].(A4.25)HereTisthesmallestconstantsuchthat(A4.20)holds.Specifically,forafunctionalf∈V
,wehavethatf=sup|f(u)|uV≤1. 404A.AppendicesForexample,inthecaseV=W=Rnandu=u,thenormofaVWlinear,boundedoperatorthatisrepresentedbyamatrixA∈Rn,ncoincideswiththecorrespondinginducedmatrixnorm(cf.AppendixA.3).Let(V,·V)beanormedspace.Asequence(ui)iinViscalledaCauchysequenceifforanyε>0thereexistsanumbern0∈Nsuchthatui−ujV≤εforalli,j∈Nwithi,j≥n0.ThespaceViscalledcompleteoraBanachspaceifforanyCauchysequence(ui)iinVthereexistsanelementu∈Vsuchthatui→ufori→∞.Ifthenorm·VofaBanachspaceVisinducedbyascalarproduct,thenViscalledaHilbertspace.AsubspaceWofaBanachspaceiscompleteiffitisclosed.Abasicprobleminthevariationaltreatmentofboundaryvalueproblemsconsistsinthefactthatthespaceofcontinuousfunctions(cf.thepreliminarydefi-nition(2.7)),whichisrequiredtobetakenasabasis,isnotcompletewithrespecttothenorm(·l,l=0orl=1).However,ifinadditiontothenormedspace(W,·),alargerspaceVisgiventhatiscompletewithrespecttothenorm·,thenthatspaceortheclosureWC:=W(A4.26)(asthesmallestBanachspacecontainingW)canbeused.Suchacom-pletioncanbeintroducedforanynormedspaceinanabstractway.Theproblemisthatthe“nature”ofthelimitingelementsremainsvague.Iftherelation(A4.26)isvalidforsomenormedspaceW,thenWiscalleddenseinW.CInfact,givenW,all“essential”elementsofWCarealreadycaptured.Forexample,ifTisalinear,continuousoperatorTfrom(W,C·)toanothernormedspace,thentheidentityT(u)=0forallu∈W(A4.27)issufficientforT(u)=0forallu∈W.C(A4.28)Thespaceoflinear,boundedoperatorsiscompleteiftheimagespaceiscomplete.Inparticular,thespaceV
oflinear,boundedfunctionalsonthenormedspaceVisalwayscomplete.A.5FunctionSpacesInthissectionG⊂Rddenotesaboundeddomain.ThefunctionspaceC(G)containsall(real-valued)functionsdefinedonGthatarecontinuousinG.ByCl(G),l∈N,thesetofl-timescontinuouslydifferentiablefunctionsonGisdenoted.Usually,forthesakeofconsistency,0∞D∞ltheconventionsC(G):=C(G)andC(G):=C(G)areused.l=0 A.5.FunctionSpaces405FunctionsfromCl(G),l∈N,andC∞(G)neednotbebounded,asfor0d=1theexamplef(x):=x−1,x∈(0,1)shows.Toovercomethisdifficulty,furtherspacesofcontinuousfunctionsareintroduced.ThespaceC(G)containsallboundedanduniformlycontin-uousfunctionsonG,whereasCl(G),l∈N,consistsoffunctionswithboundedanduniformlycontinuousderivativesuptoorderlonG.Herethe0∞D∞lconventionsC(G):=C(G)andC(G):=C(G)areused,too.l=0ThespaceC(G),respectivelyCl(G),l∈N,denotesthesetofallthose00continuous,respectivelyl-timescontinuouslydifferentiable,functions,thesupportsofwhicharecontainedinG.OftenthissetiscalledthesetoffunctionswithcompactsupportinG.SinceGisbounded,thismeansthatthesupportsdonotintersectboundarypointsofG.WealsosetC0(G):=0C(G)andC∞(G):=C(G)∩C∞(G).000ThelinearspaceLp(G),p∈[1,∞),containsallLebesguemeasurablefunctionsdefinedonGwhosepthpoweroftheirabsolutevalueisLebesgueintegrableonG.ThenorminLp(G)isdefinedasfollows:
1/ppu0,p,G:=|u|dx,p∈[1,∞).GInthecasep=2,thespecificationofpisfrequentlyomitted;thatis,u=u.TheL2(G)-scalarproduct0,G0,2,G
u,v:=uvdx,u,v∈L2(G),0,GG(inducestheL2(G)-normbysettingu:=u,u.0,G0,GThespaceL∞(G)containsallmeasurable,essentiallyboundedfunctionsonG,whereafunctionu:G→Riscalledessentiallyboundedifthequantityu∞,G:=infsup|u(x)|G0⊂G:|G0|d=0x∈GG0isfinite.Forcontinuousfunctions,thisnormcoincideswiththeusualmaximumnorm:u∞,G=max|u(x)|,u∈C(G).x∈GFor1≤q≤p≤∞,wehaveLp(G)⊂Lq(G),andtheembeddingiscontinuous.ThespaceWl(G),l∈N,p∈[1,∞],consistsofalll-timesweaklydiffer-pentiablefunctionsfromL(G)withderivativesinLp(G).Inthespecialcasepp=2,wealsowriteHl(G):=Wl(G).Inanalogytothecaseofcontinuous2functions,theconventionH0(G):=L2(G)isused.ThenorminWl(G)isp 406A.Appendicesdefinedasfollows:$
%1/pu:=|∂αu|pdx,p∈[1,∞),l,p,GG|α|≤lu:=max|∂αu|.l,∞,G∞,G|α|≤lInHl(G)ascalarproductcanbedefinedby
u,v:=∂αu∂αvdx,u,v∈Hl(G).l,GG|α|≤lThenorminducedbythisscalarproductisdenotedby·l,G,l∈N:(ul,G:=u,ul,G.Forl∈N,thesymbol|·|standsforthecorrespondingHl(G)-seminorm:l,GEF
F|u|l,G:=G|∂αu|2dx.G|α|=lThespaceH1(G)isdefinedastheclosure(orcompletion)ofC∞(G)inthe00norm·ofH1(G).1Convention:Usually,inthecaseG=Ωthespecificationofthedomainintheabovenormsandscalarproductsisomitted.Inthestudyofpartialdifferentialequations,itisoftendesirabletospeakofboundaryvaluesoffunctionsdefinedonthedomainG.Inthisrespect,theLebesguespacesoffunctionsthataresquareintegrableatthebound-aryofGareimportant.Tointroducethesespaces,somepreparationsarenecessary.
Inwhatfollows,apointx∈Rdiswrittenintheformx=xwithxdx
=(x,...,x)T∈Rd−1.1d−1AdomainG⊂Rdissaidtobelocatedatonesideof∂Gifforanyx∈∂GthereexistanopenneighbourhoodU⊂RdandanorthogonalmappingxQinRdsuchthatthepointxismappedtoapointˆx=(ˆx,...,xˆ)T,x1dandsoUismappedontoaneighbourhoodU⊂Rdofˆx,whereinthexxˆneighbourhoodUxˆthefollowingpropertieshold:(1)TheimageofUx∩∂GisthegraphofsomefunctionΨx:Yx⊂Rd−1→R;thatis,ˆx=Ψ(ˆx,...,xˆ)=Ψ(ˆx
)forˆx
∈Y.dx1d−1xx(2)TheimageofUx∩Gis“abovethisgraph”(i.e.,thepointsinUx∩Gcorrespondtoˆxd>0). A.5.FunctionSpaces407(3)TheimageofU∩(RdG)is“belowthisgraph”(i.e.,thepointsinxU∩(RdG)correspondtoˆx<0).xdAdomainGthatislocatedatonesideof∂GiscalledaCldomain,l∈N,respectivelyaLipschitz(ian)domain,ifallΨxarel-timescontinuouslydifferentiable,respectivelyLipschitzcontinuous,inYx.BoundedLipschitzdomainsarealsocalledstronglyLipschitz.Forboundeddomainslocatedatonesideof∂G,itiswellknown(cf.,e.g.[37])thatfromthewholesetofneighbourhoods{Ux}x∈∂Gtherecanbeselectedafamily{U}noffinitelymanyneighbourhoodscovering∂G,i.e.,)ii=1nn∈Nand∂G⊂i=1Ui.Furthermore,foranysuchfamilythereexistsasystemoffunctions{ϕ}nwiththepropertiesϕ∈C∞(U),ϕ(x)∈[0,1]ii=1i0iinforallx∈Uiandi=1ϕi(x)=1forallx∈∂G.Suchasystemiscalledapartitionofunity.IfthedomainGisatleastLipschitzian,thenLebesgue’sintegralovertheboundaryofGisdefinedbymeansofthosepartitionsofunity.Incor-respondencetothedefinitionofaLipschitzdomain,Qi,Ψi,andYidenotetheorthogonalmappingonUi,thefunctiondescribingthecorrespondinglocalboundary,andthepreimageofQi(Ui∩∂G)withrespecttoΨi.Afunctionv:∂G→RiscalledLebesgueintegrableover∂Gifthe

Txˆ1compositefunctionsˆx→vQiΨi(ˆx
)belongtoL(Yi).Theintegralisdefinedasfollows:
n
v(s)ds:=v(s)ϕi(s)ds∂Gi=1∂Gn


TxˆTxˆ:=vQiΨi(ˆx
)ϕiQiΨi(ˆx
)i=1Yi(×|det(∂Ψ(ˆx
)∂Ψ(ˆx
))d−1|dxˆ
.jikij,k=1Afunctionv:∂G→RbelongstoL2(∂G)iffbothvandv2areLebesgueintegrableover∂G.Intheinvestigationoftime-dependentpartialdifferentialequations,lin-earspaceswhoseelementsarefunctionsofthetimevariablet∈[0,T],T>0,withvaluesinanormedspaceXareofinterest.Afunctionv:[0,T]→Xiscalledcontinuouson[0,T]ifforallt∈[0,T]theconvergencev(t+k)−v(t)X→0ask→0holds.ThespaceC([0,T],X)=C0([0,T],X)consistsofallcontinuousfunctionsv:[0,T]→Xsuchthatsupv(t)X<∞.t∈(0,T)ThespaceCl([0,T],X),l∈N,consistsofallcontinuousfunctionsv:[0,T]→Xthathavecontinuousderivativesuptoorderlon[0,T]withthe 408A.Appendicesnorml(i)supv(t)X.t∈(0,T)i=0ThespaceLp((0,T),X)with1≤p≤∞consistsofallfunctionson(0,T)×Ωforwhichv(t,·)∈Xforanyt∈(0,T),F∈Lp(0,T)withF(t):=v(t,·).XFurthermore,vLp((0,T),X):=FLp(0,T). 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Indexadjoint,247modified,237adsorption,12artificialdiffusionmethod,373advancingfrontmethod,179,180assembling,62algorithmelement-based,66,77Arnoldi,235node-based,66CG,223asymptoticallyoptimalmethod,199multigriditeration,243nestediteration,253Banachspace,404Newton’smethod,357Banach’sfixed-pointtheorem,345algorithmicerror,200barycentriccoordinates,117anglecondition,173basisofeigenvaluesanglecriterion,184orthogonal,300anisotropic,8,139bestapproximationerror,70ansatzspace,56,67BICGSTABmethod,238nested,240bifurcation,363properties,67biharmonicequation,111approximationbilinearform,400superconvergent,193bounded,403approximationerrorestimate,139,continuous,93144definite,400forquadraturerules,160positive,400one-dimensional,137positivedefinite,400approximationproperty,250symmetric,400aquifer,7V-elliptic,93Armijo’srule,357Vh-elliptic,156Arnoldi’smethod,235block-Gauss–Seidelmethod,211algorithm,235block-Jacobimethod,211 416IndexBochnerintegral,289conjugategradient,seeCGboundary,393connectivitycondition,173boundarycondition,15conormal,16Dirichlet,15conormalderivative,98flux,15conservativeform,14homogeneous,15conservativityinhomogeneous,15discreteglobal,278mixed,15consistency,28Neumann,16consistencyerror,28,156boundarypoint,393constitutiverelationship,7boundaryvalueproblem,15continuationmethod,357,363adjoint,145continuity,402regular,145continuousproblem,21weaksolution,107approximation,21Bramble–Hilbertlemma,135contraction,402bulkdensity,12contractionnumber,199controldomain,257Cantor’sfunction,53controlvolume,257capillarypressure,10convectionCauchysequence,404forced,5,12Cauchy–Schwarzinequality,400natural,5CGmethod,221convection-diffusionequation,12algorithm,223convection-dominated,268errorreduction,224convectivepart,12withpreconditioning,228convergence,27CGNEmethod,235global,343CGNRmethod,234linear,343characteristics,388local,343Chebyshevpolynomial,225quadratic,343Choleskydecomposition,84superlinear,343incomplete,231withorderofconvergencep,343modifiedwithrespecttoanorm,401incomplete,232correction,201chordmethod,354Crank-Nicolsonmethod,313circlecriterion,184cut-offstrategy,187closure,393Cuthill–McKeemethod,89coarsegridcorrection,242,243coefficient,16Darcyvelocity,7collocationmethod,68Darcy’slaw,8collocationpoint,68decompositioncolumnsumcriterionregular,232strict,398definiteness,400comparisonprinciple,40,328degreeoffreedom,62,115,120completion,404Delaunaytriangulation,178,263complexity,88dense,96,288,404component,5density,7conditionnumber,209,397derivativespectral,398generalized,53conjugate,219material,388 Index417weak,53,289dualityargument,145diagonalfield,362diagonalscaling,230edgeswap,181diagonalswap,181eigenfunction,285differencequotient,23eigenvalue,285,291,394backward,23eigenvector,291,394forward,23element,57symmetric,23isoparametric,122,169differentialequationelementstiffnessmatrix,78convection-dominated,12,368element-nodetable,74degenerate,9ellipticityelliptic,17uniform,100homogeneous,16embedding,403khyperbolic,17H(Ω)inC(Ω),99¯inhomogeneous,16emptyspherecriterion,178linear,16energynorm,218nonlinear,16energynormestimates,132order,16energyscalarproduct,217parabolic,17equidistributionstrategy,187quasilinear,16error,201semilinear,16,360errorequation,68,242typeof,17errorestimatedifferentialequationmodelapriori,131,185instationary,8anisotropic,144linear,8errorestimatorstationary,8aposteriori,186diffusion,5asymptoticallyexact,187diffusivemassflux,11efficient,186diffusivepart,12reliable,186Dirichletdomain,262residual,188Dirichletproblemdual-weighted,194solvability,104robust,187discreteproblem,21errorleveldiscretization,21relative,199five-pointstencil,24Eulermethodupwind,372explicit,313discretizationapproach,55implicit,313discretizationparameter,21extensivequantity,7divergence,20extrapolationfactor,215divergenceform,14extrapolationmethod,215domain,19,394lC,407face,123kC-,96familyoftriangulations∞C-,96quasi-uniform,165Lipschitz,96,407regular,138strongly,407Fick’slaw,11domainof(absolute)stability,317fill-in,85Donalddiagram,265finitedifferencemethod,17,24dualproblem,194finiteelement,115,116 418Index1C-,115,127Lebesgueintegrable,407affineequivalent,122measurable,393Bogner–Fox–Schmitrectangle,127piecewisecontinuous,480C-,115support,394cubicansatzonsimplex,121functional,403cubicHermiteansatzonsimplex,functionalmatrix,348126functionsd-polynomialansatzoncuboid,123equalalmosteverywhere,393equivalent,122Hermite,126Galerkinmethod,56Lagrange,115,126stability,69linear,57uniquesolvability,63linearansatzonsimplex,119Galerkinproduct,248quadraticansatzonsimplex,120Galerkin/leastsquares–FEM,377simplicial,117Gauss’sdivergencetheorem,14,47,finiteelementcode266assembling,176Gauss–Seidelmethod,204kernel,176convergence,204,205post-processor,176symmetric,211finiteelementdiscretizationGaussianelimination,82conforming,114generatingfunction,316condition,115GMRESmethod,235nonconforming,114truncated,238finiteelementmethod,18withrestart,238characterization,67gradient,20convergencerate,131gradientmethod,218maximumprinciple,175errorreduction,219mortar,180gradientrecovery,192finitevolumemethod,18graphcell-centred,258dual,263cell-vertex,258gridnode-centred,258chimera,180semidiscrete,297combined,180five-pointstencil,24hierarchicallystructured,180fixedpoint,342logicallystructured,177fixed-pointiteration,200,344overset,180consistent,200structured,176convergencetheorem,201inthestrictsense,176fluid,5inthewidersense,177Fouriercoefficient,287unstructured,177Fourierexpansion,287gridadaptation,187Friedrichs–Kellertriangulation,64gridcoarsening,183frontalmethod,87gridfunction,24fulldiscretization,293gridpoint,21,22fullupwindmethod,373closetotheboundary,24,327functionfarfromtheboundary,24,327almosteverywherevanishing,393neighbour,23continuous,407essentiallybounded,405harmonic,31 Index419heatequation,9L0-matrix,399Hermiteelement,126L-matrix,399Hessenbergmatrix,398Lagrangeelement,115,126Hilbertspace,404Lagrange–Galerkinmethod,387homogenization,6Lagrangiancoordinate,387hydraulicconductivity,8Lanczosbiorthogonalization,238Langmuirmodel,12ICfactorization,231Laplaceequation,9ill-posedness,16Laplaceoperator,20ILUfactorization,231lemmaexistence,232Bramble–Hilbert,135ILUiteration,231C´ea’s,70inequalityfirstofStrang,155ofKantorovich,218lexicographic,25Friedrichs’,105linearconvergence,199inverse,376Lipschitzconstant,402ofPoincar´e,71Lipschitzcontinuity,402inflowboundary,108loadvector,62inhomogeneity,15LUfactorization,82initialcondition,15incomplete,231initial-boundaryvalueproblem,15innerproductM-matrix,41,3991onH(Ω),54macroscale,6integralform,14mappingintegrationbyparts,97bounded,402interior,394continuous,402interpolationcontractive,402local,58linear,402interpolationerrorestimate,138,144Lipschitzcontinuous,402one-dimensional,136massactionlaw,11interpolationoperator,132massaveragemixturevelocity,7interpolationproblemmasslumping,314,365local,120massmatrix,163,296,298isotropic,8masssourcedensity,7iterationmatrixinner,355band,84outer,355bandwidth,84iterationmatrix,200consistentlyordered,213iterativemethod,342Hessenberg,398hull,84Jacobimatrix,348inversemonotone,41Jacobi’smethod,203irreducible,399convergence,204,205L0-,399jump,189L-,399jumpcondition,14LUfactorizable,82M-,399Krylov(sub)space,222monotone,399Krylovsubspaceofmonotonetype,399method,223,233pattern,231 420Indexpositivedefinite,394finitedifference,24profile,84fullupwind,373reducible,399Galerkin,56rowbandwidth,84Gauss–Seidel,204rowdiagonallydominantGMRES,235strictly,398iterative,342weakly,399Jacobi’s,203sparse,25,82,198Krylovsubspace,223,233symmetric,394Lagrange–Galerkin,387triangularlinearstationary,200lower,398mehrstellen,30upper,398movingfront,179matrixnormmultiblock,180compatible,396multigrid,243induced,397Newton’s,349mutuallyconsistent,396ofbisection,182submultiplicative,396stagenumberof,182subordinate,397one-step,316matrixpolynomial,394one-step-theta,312matrix-dependent,248overlay,177max-min-angleproperty,179PCG,228,229maximumanglecondition,144r-,181maximumcolumnsum,396relaxation,207maximumprincipleRichardson,206strong,36,39,329Ritz,56weak,36,39,329Rothe’s,294maximumrowsum,396semi-iterative,215mechanicaldispersion,11SOR,210meshwidth,21SSOR,211methodstreamlineupwindPetrov–advancingfront,179,180Galerkin,375algebraicmultigrid,240streamline-diffusion,377Arnoldi’s,235methodofconjugatedirections,219artificialdiffusion,373methodoflinesasymptoticallyoptimal,199horizontal,294BICGSTAB,238vertical,293block-Gauss–Seidel,211methodofsimultaneousblock-Jacobi,211displacements,203CG,221methodofsuccessivedisplacements,classicalRitz–Galerkin,67204collocation,68MICdecomposition,232consistent,28microscale,5convergence,27minimumanglecondition,141Crank-Nicolson,313minimumprinciple,36Cuthill–McKee,89mobility,10reverse,90moleculardiffusivity,11Eulerexplicit,313monotonicityEulerimplicit,313inverse,41,280extrapolation,215monotonicitytest,357 Index421movingfrontmethod,179equivalent,395multi-index,53,394numberinglength,53,394columnwise,25order,53,394rowwise,25multiblockmethod,180multigriditeration,243octreetechnique,177algorithm,243one-stepmethod,316multigridmethod,243A-stable,317algebraic,240strongly,319L-stable,319neighbour,38nonexpansive,316nestediteration,200,252stable,320algorithm,253one-step-thetamethod,312Neumann’slemma,398operator,403Newton’smethod,349operatornorm,403algorithm,357orderofconsistency,28damped,357orderofconvergence,27inexact,355orthogonal,401simplified,353orthogonalityoftheerror,68nodalbasis,61,125outerunitnormal,14,97nodalvalue,58outflowboundary,108node,57,115overlaymethod,177adjacent,127overrelaxation,209degree,89overshooting,371neighbour,63,89,211norm,400parabolicboundary,3252discreteL-,27parallelogramidentity,400equivalenceof,401Parseval’sidentity,292Euclidean,395particlevelocity,7Frobenius,396partition,256inducedbyascalarproduct,400partitionofunity,407p-,395PCGmatrix,395method,228,229maximum,395P´ecletnumbermaximum,27global,12,368maximumcolumnsum,396grid,372maximumrowsum,396local,269ofanoperator,403permeability,8spectral,397perturbationlemma,398streamline-diffusion,378phase,5stronger,401immiscible,7total,396phaseaveragevector,395extrinsic,6ε-weighted,374intrinsic,6normalderivative,98k-phaseflow,5normalequations,234(k+1)-phasesystem,5normedspacepiezometrichead,8complete,404pointnormsboundary,40 422Indexclosetotheboundary,40relativepermeability,9farfromtheboundary,40relaxationmethod,207Poissonequation,8relaxationparameter,207Dirichletproblem,19representativeelementaryvolume,6polynomialresidual,188,189,201,244characteristic,395inner,355matrix,394restriction,248porescale,5canonical,247porespace,5Richardsequation,10porosity,6Richardsonmethod,206porousmedium,5optimalrelaxationparameter,208porousmediumequation,9Ritzmethod,56preconditioner,227Ritzprojection,304preconditioning,207,227Ritz–Galerkinmethodfromtheleft,227classical,67fromtheright,227rootofequation,342preprocessor,176Rothe’smethod,294pressurerowsumcriterionglobal,10strict,204,398principleofvirtualwork,49weak,205,399projection2:1-rule,181elliptic,303,304prolongation,246,247saturated,10canonical,246saturated-unsaturatedflow,10pyramidalfunction,62saturation,7saturationconcentration,12quadraturepoints,80scalarproduct,400quadraturerule,80,151energy,217accuracy,152Euclidean,401Gauss–(Legendre),153semi-iterativemethod,215integrationpoints,151semidiscreteproblem,295nodal,152semidiscretization,293trapezoidalrule,66,80,153seminorm,400,406weights,151separationofvariables,285quadtreetechnique,177setclosed,393,402range,343connected,394reactionconvex,394homogeneous,13open,394inhomogeneous,11setofmeasurezero,393surface,11shapefunction,59recoveryoperator,193cubicansatzonsimplex,121redmblackordering,212d-polynomialansatzoncube,123reductionstrategy,187linearansatzonsimplex,120referenceelement,58quadraticansatzonsimplex,121standardsimplicial,117simplexrefinementbarycentre,119iterative,231degenerate,117red/green,181face,117 Index423regulard-,117superpositionprinciple,16sliverelement,179surfacecoordinate,119smoothingsystemofequationsbarycentric,181positivereal,233Laplacian,181weightedbarycentric,181testfunction,47smoothingproperty,239,250theoremsmoothingstep,178,242ofAubinandNitsche,145aposteriori,243ofKahan,212apriori,243ofLax–Milgram,93smoothnessrequirements,20ofOstrowskiandReich,212Sobolevspace,54,94ofPoincar´e,71solidmatrix,5Trace,96soluteconcentration,11Thiessenpolygon,262solutionthree-termrecursion,234classical,21timelevel,312ofan(initial-)boundaryvaluetimestep,312problem,17tortuosityfactor,11variational,49trace,97weak,49,290transformationuniqueness,51compatible,134solvent,5isoparametric,168SORmethod,210,213transformationformula,137convergence,212transmissioncondition,34optimalrelaxationparameter,213triangleinequality,400sorbedconcentration,12triangulation,56,114sourceterm,14anisotropic,140spaceconforming,56,125normed,400element,114space-timecylinder,15properties,114bottom,15refinement,76lateralsurface,15truncationerror,28spectralnorm,397two-griditeration,242spectralradius,395algorithm,242spectrum,395splitpreconditioning,228underrelaxation,209SSORmethod,211unsaturated,10stabilityfunction,316upscaling,6stabilityproperties,36upwinddiscretization,372stable,28upwindingstaticcondensation,128exponential,269stationarypoint,217full,269stepsize,21stiffnessmatrix,62,296,298V-cycle,244elemententries,76V-elliptic,69streamlineupwindPetrov–Galerkinvariationofconstants,286method,375variationalequation,49streamline-diffusionmethod,377equivalencetominimizationstreamline-diffusionnorm,378problem,50 424Indexsolvability,93regular,262viscosity,8volumeaveraging,6W-cycle,244volumetricfluidvelocity,7waterpressure,8volumetricwatercontent,11weight,30,80Voronoidiagram,262well-posedness,16Voronoipolygon,262Wigner–Seitzcell,262Voronoitesselation,1782Voronoivertex,262Z–estimate,192degenerate,262zerooffunctionf,342