Analysis of post--processing for nonconforming fin

AnalysisofPost-ProcessingforNonconformingFiniteElementSolutionsF.SchieweckInstitutfurAnalysisundNumerikOtto-von-Guericke-UniversitatMagdeburgPostfach4120,D-39016Magdeburg,GermanyAbstractForageneralpairofagivennonconformingandadesiredconformingniteel-ementspace,weconstructandanalyzeatransferoperatorthatcanbeusedforapost-processingprocedurewhereaconformingapproximationiscomputedfromanonconformingniteelementsolution.Forthispost-processing,weprovelocalandglobalestimatesoftheerrorbetweentheexactsolutionandtheconformingapprox-imationofthenonconformingdiscretesolution.Ouranalysisisindependentoftheconcreteproblemthathastobesolved.Applicationsofthisanalysisarediscussed,inparticularintheeldofaposteriorierrorcontrol.1991MathematicsSubjectClassication(MSC):35A40,65N15,65N30,65D05,65D15Keywords:nonconformingandconformingniteelements,post-processing,erroresti-mates,aposteriorierrorindicator1IntroductionFormathematicalreasons,itissometimesusefultoweakenthecontinuityrequirementofniteelementapproximationsandtoworkwithnonconformingniteelements.AnexampleistheStokesorNavier-Stokesproblemwherestandardlow-orderconformingelementpairsdonotsatisfytheBabuska-Brezzistabilitycondition(see[2]).HerethenonconformingP1=P0triangularelementpairofCrouzeixandRaviart(see[4])satisesthatconditionandprovidesanoptimalorderofconvergence.ThesameistruefortheanalogueofthiselementpairinthecaseofquadrilateralelementswhichhasbeenproposedbyRannacherandTurek(see[7])andwhichhasprovedtobeattractiveforthenumericalsolutionoftheNavier-Stokesequations(see[8],[9],[10]).Moreover,anadvantageofthesenonconformingniteesthelocalcommunicationforaparallelizationofthemethod,particularlyinthe3Dcase.However,forpracticalreasons(e.g.graphicalpost-processing),onewantstohaveaniteelementapproximationofthesolutionwhichis\niceinsomesense,i.e.whichisatleastcontinuous.Theobviousideausedinpracticeistocomputeinapost-processingstepfromasolutionuNhinagivennonconformingniteelementspaceVNhanapproximationuChinsomedesiredconformingniteelementspaceVCh.Inthispaper,wedeneageneraltransferoperatorRh:VNh!VChforanarbitrarypairofniteelementspacesVNhandVCh.Thisoperatorisbasedonaveragingoflocalnodalfunctionals.Thecomputationofthe\smoothedfunctionuCh=RhuNhcanbeimplementedecientlybyoneloopovertheelementsfollowedbyoneloopoverthenodescorrespondingtothespaceVCh.WeprovelocalandglobalestimatesfortheerrorbetweentheexactsolutionuandtheconformingapproximationRhuNhintheL2-normandtheH1-semi-norm.TheseestimatesshowthatthiserroriscontrolledbytheerroroftheusednonconformingniteelementmethoduuNhandtheinterpolationerroroftheexactsolutionubymeansofaconformingniteelementfunctionvCh2VCh.One\nicefeatureofouranalysisisthatitisindependentoftheconcreteproblemwhichshouldbesolvedbytheniteelementmethod.TheonlyfactthatwehavetoassumeisthatourcomputednonconformingniteelementsolutionuNh2VNhisclosetoasucientlysmoothsolutionuofthecontinuousproblem.ThecomputablequantitiespostK,beingdenedasthelocalnormsofthe\post-processingerroruNhRhuNhontheelementsK,canbeusedforaposteriorierrorcontrol.Foramodelproblem,weclaimanaposteriorierrorestimatefortheerroruRhuNhinadiscreteH1-semi-normwhichcontainsthequantitiespostK.Thisassertionwillbeprovedinaforthcomingpaper.2PreliminariesandNotationsLetIRd,d2f2;3g,beaboundedpolyhedraldomainandTharegulardecompositionofintoelementsK2Th.ThediameterofanelementKisdenotedbyhKandtheelementsareassumedtobeshaperegularintheusualsense(seee.g.[1]).Letusnotethatthisassumptionincludesthecaseofalocallyrenedgridwheretheratioh=hKwithh:=maxK2ThhKmaybeverylargeforelementsKinsomecriticalsubregionsof.WeconsiderthreeniteelementspacesdenedonthedecompositionTh,agivennoncon-formingspaceVNh,somedesiredconformingspaceVChandanauxiliaryspaceVAh.Forthedenitionofthesespaces,weusethereferencetransformationFK:bK!KthatmapsaxedreferenceelementbKontotheactualelementK2Th.Weassume,forsimplicity,thatwehaveonlyonereferenceelementbK.However,ouranalysiscanbegeneralizedtothesituationthatwehaveaxednitesetofreferenceelementswhichoccursforexampleifthedecompositionofcontainsbothtriangularandquadrilateralelements.2Inthefollowing,foragivendomainGIRd,wedenotebyPm(G)thesetofallpolynomialsonGwithdegreelessorequaltom.LetuscharacterizethenonconformingspaceVNhasasubspaceofacorrespondingdiscon-tinuousspaceeVNh,i.e.VNheVNh:=v2L2():vK2PNK8K2Th;(1)wherePNK:=np=bpF1K:bp2bPNowithP0(bK)bPNPr1(bK)(2)isthespaceoflocalshapefunctionsofVNhonelementKdenedbysomegivenreferencespacebPN.Afunctionvh2eVNhbelongstoVNhifitiscontinuouswithrespecttothosenodalfunctionalsthatareassociatedwithVNh.Similarly,lettheconformingspaceVChbecharacterizedasVCheVCh:=v2C0():vK2PCK8K2Th;(3)wherePCKisagaindenedviatransformationofsomegivenreferencespacebPC,i.e.PCK:=np=bpF1K:bp2bPCowithP0(bK)bPCPr2(bK):(4)Afunctionvh2eVChbelongstoVChifitsatisessomedesiredboundaryconditions.ThecontinuityofthepiecewisesmoothfunctionsinVChi