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1、Numer.Math.63,183-194(1992)NumerischeMathematik~Springer-Verlag1992AnalysisofmixedfiniteelementmethodsonlocallyrefinedgridsRichardE.EwingandJunPingWangDepartmentofMathematics,CollegeofArtsandSciences,UniversityofWyoming,P.O.Box3036,Laramie,WY82071-3036,USAReceivedJanuary30,1991Summary.Weco
2、nsiderthemixedfiniteelementmethodforlocallyrefinedtrian-gulations.Alocalprojectionoperator/)hisdefinedtosatisfythecommutationpropertythatisrequiredinthegeneraltheoryofmixedmethods.Ourresultscanbeappliedtoeveryknownspaceofarbitraryorderoverrectanglesortriangles.Optimal-ordererrorestimatesan
3、dsuperconvergenceforthefluxalongtheGausslinesareestablished.MathematicsSubjectClassification(I991):65N30I.IntroductionRecently,greatattentionhasbeenfocusedonthefiniteelementmethodassociat-edwithlocallyrefinedgrids(cf.[2,7,11,16]).Itisbelievedthatthetechniqueoflocalrefinementcanprovidenoton
4、lyagoodapproximatesolutionbutalsoasimpleralgebraicequationwhendomaindecompositionandmultileveltech-niquesareapplied,especiallyforproblemswithroughsolutions.Ourobjectiveinthispaperistostudytheconvergencebehaviorofthislocalrefinementtech-niqueappliedtomixedfiniteelementmethods.Forsimplicity,
5、wetakeasourmodelthesecond-orderellipticequationwithahomogeneousNeumannbound-arycondition(1.1)-V.(aVp)=fin(2,aVp.v=O,ont~swherea(x,y)isadiagonalmatrixwhichisuniformlypositivedefinitein(x,y)~sandvtheunitoutwardnormalvectortotheboundary0(2.HereVandV.denotethegradientandthedivergenceoperators,
6、respectively.AssumethatQisanopenboundedpolygonaldomaininFxz.Theresultscaneasilybeextendedtoproblemswithdomain(2havingacurvedboundaryortoaboundedopensubsetofl~,3.Correspondenceto:J.Wang184R.E.EwingandJ.P.WangTherefluxoftheproblem(1.1)isdefinedby(1.2)u=-a(x,y)Vp;itisaquantitythatisoftenofphy
7、sicalinterest.Followingmanyfluidflowapplications,thefunctionpwillbecalledthe'pressure'inthispaper.LetH(div,f2)bethespaceofvector-valuedL2(squareintegrable)functionssothatthedivergenceisalsosquaresummableoverf2.ThespaceH(div,f2)isequippedwiththenaturalnormIIvlI
