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1、VidarThoméeGalerkinFiniteElementMethodsforParabolicProblemsABCPrefaceMypurposeinthismonographistopresentanessentiallyself-containedaccountofthemathematicaltheoryofGalerkinfiniteelementmethodsasappliedtoparabolicpartialdifferentialequations.Theemphasesan
2、dselectionoftopicsreflectsmyowninvolvementinthefieldoverthepast25years,andmyambitionhasbeentostressideasandmethodsofanalysisratherthantodescribethemostgeneralandfarreachingresultspossible.SincetheformulationandanalysisofGalerkinfiniteelementmethodsforpar
3、abolicproblemsaregenerallybasedonideasandresultsfromthecorrespondingtheoryforstationaryellipticproblems,suchmaterialisoftenincludedinthepresentation.ThebasisofthisworkismyearliertextentitledGalerkinFiniteElementMethodsforParabolicProblems,SpringerLect
4、ureNotesinMathematics,No.1054,from1984.Thishasbeenoutofprintforseveralyears,andIhavefeltaneedandbeenencouragedbycolleaguesandfriendstopublishanupdatedversion.IndoingsoIhaveincludedmostofthecontentsofthe14chaptersoftheearlierworkinanupdatedandrevisedfo
5、rm,andaddedfournewchapters,onsemigroupmethods,onmultistepschemes,onincompleteiterativesolutionofthelinearalgebraicsystemsatthetimelevels,andonsemilinearequations.Theoldchaptersonfullydiscretemethodshavebeenreworkedbyfirsttreatingthetimediscretizationof
6、anabstractdifferentialequationinaHilbertspacesetting,andthechapteronthediscontinuousGalerkinmethodhasbeencompletelyrewritten.Thefollowingisanoutlineofthecontentsofthebook:IntheintroductoryChapter1webeginwithareviewofstandardmaterialonthefiniteelementmet
7、hodforDirichlet’sproblemforPoisson’sequationinaboundeddomain,andconsiderthenthesimplestGalerkinfiniteelementmethodsforthecorrespondinginitial-boundaryvalueproblemforthelinearheatequation.Thediscretemethodsarebasedonassociatedweak,orvari-ational,formula
8、tionsoftheproblemsandemployfirstpiecewiselinearandthenmoregeneralapproximatingfunctionswhichvanishontheboundaryofthedomain.Forthesemodelproblemswedemonstratethebasicerrorestimatesinenergyandmeansquarenorms,intheparaboliccasefirstforthesemidiscre