The equilibrium method for local, a posteriori, po

TheEquilibriummethodforLocal,APosteriori,PointwiseEstimationoftheFiniteElementErrorJensHuggeryInstituteforMathematicsUniversityofCopenhagen,DenmarkSeptember19,1995AbstractTheequilibriummethodfortherecoveryofniteelementerrorsisextendedandgiventhetheoreticalfoundationnecessarytoemployittothepointwiserecoveryoftheniteelementerrorfunction.Thepracticalimplicationsoftheresultingtheoryisstudiedextensivelythroughacomputationalexample.ResearchsupportedbyIAN/CNR(IstitutoDiAnalisiNumerica-CNR)grant211.01.26,andSNF(TheDanishNaturalScienceResearchCouncil)grant11-9030ye-mail:[e-mail]11IntroductionTheequilibrationmethodforndingerrorestimatorsfortheniteelementmethodwasintroducedbyLadevezeinhisdoctoralthesis.Amoreaccessiblereferenceis[1].LatertheapproachwasappliedforexampleinalocalsettingbyBankandWeiserin[2]andinaglobalsettingbyKellyin[3],andmostrecentlyinalocalsettingbyOdenandAinsworthin[4],[5]and[6].Commonintheliteraturetodatehasbeenanapproachthatinafewwordscanbedescribedby\Firstpostulateanerrorestimator(possiblyjustiedbyphysicalarguments);thenprovethattheestimatederrorisclose(upperand/orlowerboundinanapriorigivennorm,generallytheenergynorm)totheexacterrorundercertainregularityassumptions.In[7]and[8]Huggerpresentsamethodforthepointwiserecoveryoftheerror,leavingthepossibilityopenforpostprocessingoftheniteelementsolutionorfortheestimationoftheniteelementerrorinanarbitrarynorm.Therstarticlepresentsthesolutionintheonedimensionalcasewhereeverythingworksverywell.Thesecondarticledealswiththetwodimensionalcaseandtheresultspresentedarenotcomplete,sincetheissueofhowtoapproximatetheerrorontheedgesoftheniteelementsisresolvedonlyinthecoarsestmanner.Thestandardequilibrationapproachisnotdirectlyapplicablesincetheproofsforequivalenceoftheestimatedandexacterrorsdoesnotexistforthepointwiseerror.(TheC1normisnotavailable).Inthisarticletheideaoftheequilibrationofthejumpoftheerroracrosselementboundariesisputinatheoreticalsettingthatallowstheimmediateapplicationofthetheoryofpointwiseerrorestimationfrom[8].Theaimisrstofalltosolvetheproblemofprovidinghighqualitylocalboundaryconditionsontheelementedgesforthepointwiseestimationoftheerror.Asecondgoalhasbeentoprovideacleartheoryfortheequilibrationapproachforerrorestimationwheretheestimatederrorisrecoveredbysolvingasetofequationsarisingbyclearlyexplainedsimplicationsofexactequationsfortheerror.Therebywehaveexactknowledgeabouttheerrorscommittedintheerrorestimationprocess.Thismayeventuallyleadtoimprovementsinthestandardequilibrationmethod,andalsoreplacestheequivalenceresultsmentionedabovethatcangenerallyonlybeprovedeasilyforsimplelinearboundaryvalueproblemsusingtheenergynormorarelatednorm.Insections2,3,and4classicalandvariationalformulationsaregiven,resultinginanexpressionfortheerrorestimatordependingonsocalledequilibriumfunctionsand.Insection5themethodforpointwiserecovery2oftheequilibriumfunctionsbasedonthework[6]byAinsworthandOdenispresentedandsetintheframeworkdevelopedinsections2{4.Section6introducesthetestproblemtobeusedfortheinvestigationofthepracticalperformanceofthemethod.Theninsection7thepointwiserecoveryoftheequilibriumfunctionsisstudiedforthetestproblem,andinsection8theresultingerrorestimatorisevaluated.Finallysection9concludesthepaper.ItshouldbenotedthatallcomputationsandgraphicsforthearticlehavebeenperformedwiththesymbolicmanipulationpackageMapleV.3.Ithasbeenofsecondaryinteresttoevaluatetheperformanceofsuchasymbolicmanipulation/graphicspackagewithincomputationalnumericalanalysis.2Localboundaryvalueproblemsfortheer-rorLetCk,L2,andH1betheusualspacesofktimescontinuouslydierentiablefunctions,squareintegrablefunctions,andweaklydierentiablefunctionswithfunctionsandweakrstderivativesinL2,respectively.LetadomainR2andaniteelementmeshT=fkgNk=1overbegiven.Letf@1;@[e-mail]atedsolutionstothefollowingboundaryvalueproblem:Finduex2C2()\C0():Auex=fin;(1)uex=0on@1;anda(uex)n=gon@2;Au=ra(u)+b(u):Herenisanoutwardnormalvectortotheboundaryinquestion,anda,b,f,andgaresuitablefunctions.aandbdependonuandrubutonnohigherderivativesofu.Alsoa;b2C2(),f2L2(),andg2L2(@2),butthedependenceonx2issuppressedtosimplifynotation.Letufe2C0(),ufe=0on@1,beaniteelement,piecewisepolynomialapproximationtoasolutionuextoproblem(1).Denetheerrorfunctionebye=uexufe:(2)Weshallstudylocal,linearproblemsfortheerror,andthereforestartlin-earizingaandbaroundufe,deningtheanelinearizations^aand^bby^a(u)=a(ufe)+aL(uufe)’a(u);(3)3^b(u)=b(ufe)+bL(uufe)’b(u):(4)HereaLandbLaretheFrechetderivativesinthepointufeofaandbre-spectively.Notethattheyarelinear,andthatifaandbarelinearthen^a=aL=a,^b=bL=b,and\’inexpressions(3)and(4)arereplacedby\=.Denealsotheaneandthelineardierentialoperators^AandALby^Au=r^a(u)+^b(u)’Au;(5)ALu=raL(u)+bL(u)’Au:(6)Noteagainthatifaandbarelinear,\’in(5{6)arereplacedwith\=.Letnallyforanyfunctionudenedon,andforanyk2f1;:::;Ng,[e-mail]