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1、MATHEMATICSOFCOMPUTATIONS0025-5718(2011)02524-XArticleelectronicallypublishedonMay11,2011ANALYSISOFANADAPTIVEUZAWAFINITEELEMENTMETHODFORTHENONLINEARSTOKESPROBLEMCHRISTIANKREUZERAbstract.WedesignandstudyanadaptivealgorithmforthenumericalsolutionofthestationarynonlinearStokesprob
2、lem.Thealgorithmcanbeinterpretedasadisturbedsteepestdescentmethod,whichgeneralizesUzawa’smethodtothenonlinearcase.Theouteriterationforthepressureisadescentmethodwithfixedstep-size.TheinneriterationforthevelocityconsistsofanapproximatesolutionofanonlinearLaplaceequation,whichisre
3、alizedwithadaptivelinearfiniteelements.Thedescentdirectionismotivatedbythequasi-normwhichnaturallyarisesasdistancebetweenvelocities.Weestablishtheconvergenceofthealgorithmwithintheframeworkofdescentdirectionmethods.1.IntroductionPartialdifferentialequationslikethestationaryStokes
4、problemariseinnumerousphysicalmodels,particularlyinthemodelingofQuasi-Newtonianfluids.ForΩ⊂Rdbeingaboundedpolyhedraldomainandagivenexternalbodyforcef:Ω→Rd,thevelocityu:Ω→Rdofthefluidanditspressurep:Ω→RcanbedescribedbythestationaryStokesequations:−divA(Eu)+∇p=finΩ,(1.1)divu=0inΩ,u
5、=0on∂Ω.ThenonlineartensorA:Rd×d→Rd×dmimicsthechangeoftheviscosityofthefluidwithrespecttotheshearrateEu:=1(∇u+∇uT).Typicalchoicesin2engineeringare,amongothers,thepowerlawandtheCarreaulawr−222r−2(1.2)A(E)=ν0
6、E
7、EandA(E)=ν∞+(ν0−ν∞)(κ+
8、E
9、)2E,E∈Rd×d,wherer∈(1,∞),ν>ν≥0andκ≥0.0∞Inordert
10、otreatasmanymodelsaspossible,aswellasthecases12simultaneously,weformulate(1.1)intermsofso-calledN-functionsφ.BasedonthesefunctionswecandefinefunctionspacesVforthevelocityandQforthepressureinordertogetanappropriateweakformulationof(1.1).Thestandardfiniteelementapproachist
11、ouseanadequatepairofdiscretefunctionspacesV(T),Q(T)andthencomputetheRitzGalerkinapproximation(U,P)∈V(T)×Q(T).ReceivedbytheeditorDecember16,2009and,inrevisedform,January22,2011.2010MathematicsSubjectClassification.Primary65N30,65N12,35J60.Keywordsandphrases.Convergence,adaptivefin
12、iteelements,p-Stokes,p-Laplacian,quasinorm,Uzawaalgori
