Superconvergence of the local discontinuous Galerk

SUPERCONVERGENCEOFTHELOCALDISCONTINUOUSGALERKINMETHODFORELLIPTICPROBLEMSONCARTESIANGRIDSBERNARDOCOCKBURN,GUIDOKANSCHATy,ILARIAPERUGIAz,ANDDOMINIKSCHOTZAUxApril27,2000Abstract.Inthispaper,wepresentasuper-convergenceresultfortheLocalDiscontinuousGalerkinmethodforamodelellipticproblemonCartesiangrids.WeidentifyaspecialnumericaluxforwhichtheL2-normofthegradientandtheL2-normofthepotentialareoforderk+1=2andk+1,respectively,whentensorproductpolynomialsofdegreeatmostkareused;forarbitrarymeshes,thisspecialLDGmethodgivesonlytheordersofconvergenceofkandk+1=2,respectively.Wepresentaseriesofnumericalexampleswhichestablishthesharpnessofourtheoreticalresults.Keywords.Finiteelements,discontinuousGalerkinmethods,super-convergence,ellipticprob-lems,Cartesiangrids.AMSsubjectclassications.Primary65N30.1.Introduction.Inthispaper,wederiveapriorierrorestimatesoftheLocalDiscontinuousGalerkin(LDG)methodonCartesiangridsforthefollowingclassicalmodelellipticproblem:u=fin;u=gDonD;@u@n=gNnonN;(1.1)whereisaboundeddomainofRdandnistheoutwardunitnormaltoitsboundary=D[N;weassumethatthe(d1)-measureofDisnon-zero.Recently,Castillo,Cockburn,PerugiaandSchotzau[3]obtainedtherstapriorierroranalysisoftheLDGmethodforpurelyellipticproblems.MeshesconsistingofelementsofvariousshapesandwithhangingnodeswereconsideredandgeneralSchoolofMathematics,UniversityofMinnesota,VincentHall,Minneapolis,MN55455([e-mail]).SupportedinpartbytheNationalScienceFoundation(GrantDMS-9807491)andbytheUniversityofMinnesotaSupercomputingInstitute.yInstitutfurAngewandteMathematik,UniversitatHeidelberg,INF293/294,69120Heidelberg,Germany([e-mail]).ThisworkwassupportedinpartbytheARODAAG55-98-1-0335andbytheUniversityofMinnesotaSupercomputingInstitute.ItwascarriedoutwhentheauthorwasaVisitingProfessorattheSchoolofMathematics,UniversityofMinnesota.zDipartimentodiMatematica,UniversitadiPavia,ViaFerrata1,27100Pavia,Italy([e-mail]).SupportedinpartbytheConsiglioNazionaledelleRicerche.ThisworkwascarriedoutwhentheauthorwasaVisitingProfessorattheSchoolofMathematics,UniversityofMinnesota.xSchoolofMathematics,UniversityofMinnesota,VincentHall,Minneapolis,MN55455([e-mail]).SupportedbytheSwissNationalScienceFoundation(SchweizerischerNationalfonds).12B.Cockburn,G.Kanschat,I.PerugiaandD.Schotzaunumericaluxeswerestudied.Itwasshownthat,forverysmoothsolutions,theordersofconvergenceoftheL2-normsoftheerrorsinruandinuarekandk+1=2,respectivelywhenpolynomialsofdegreeatmostkareused.Ontheotherhand,Castillo[2]andCastillo,Cockburn,SchotzauandSchwab[4]provedthat,forone-spacedimensiontransientconvection-diusionproblems,theorderofconvergenceoftheerrorintheenergynormisoptimal,thatis,k+1,providedthattheso-callednumericaluxesaresuitablychosen.Inthispaper,weextendtheseresultstotheLDGmethodonCartesiangridsforthemulti-dimensionalellipticmodelproblem(1.1);weshowthattheordersofconvergenceintheL2-normoftheerrorinruanduarek+1=2andk+1,respectively,whentensorproductpolynomialsofdegreeatleastkareused.Ourproofofthissuper-convergenceresultisamodicationoftheanalysiscarriedoutin[3];ittakesadvantageoftheCartesianstructureofthegridandmakesuseofakeyideaintroducedbyLeSaintandRaviart[10]intheirstudyoftheoriginalDGmethodforsteady-statelineartransport.Sinceouranalysisisaspecialmodicationofthatof[3],inordertoavoidunnecessaryrepetitions,wereferthereaderto[3]foramoredetaileddescriptionoftheframeworkofourerroranalysis.Theorganizationofthispaperisasfollows.InSection2,webrieydisplaytheLDGmethodincompactform,introducethespecialnumericaluxonCartesiangridsandpresentanddiscussourmainresult.InSection3,thedetailedproofsaregivenandinSection4,wepresentseveralnumericalexperimentsshowingtheoptimalityofourtheoreticalresults.WeendinSection5withsomeconcludingremarks.2.Themainresults.InthissectionwerecalltheformulationoftheLDGmethodandidentifythespecialnumericaluxwearegoingtoinvestigateonCartesiangrids.Thenwestateanddiscussourmainresults.Aspointedoutintheintroduction,wereferto[3]formoredetailsconcerningtheformulationoftheLDGmethod.2.1.TheLDGmethod.WeassumethattheproblemdomaincanbecoveredbyaCartesiangrid.TodenetheLDGmethod,werewriteourellipticmodelproblem(1.1)asthefollowingsystemofrst-orderequations:q=ruin;(2.1)rq=fin;(2.2)u=gDonD;(2.3)qn=gNnonN:(2.4)Next,wediscretizetheaboveproblemonaCartesiangridT.ToobtaintheweakformulationwithwhichtheLDGisdened,wemultiplyequations(2.1)and(2.2)byarbitrary,smoothtestfunctionsrandv,respectively,andintegratebypartsoverthed-dimensionalrectangleK2T.Thenwereplacetheexactsolution(q;u)byitsapproximation(qN;uN)intheniteelementspaceMNVN,whereMN:=fq2(L2())d:qK2S(K)d;8K2Tg;(2.5)VN:=fu2L2():uK2S(K);8K2Tg;(2.6)andS(K):=Qk(K)=fpolynomialsofdegreeatmostkineachvariableonKg:TheLDGmethodonCartesiangridsforellipticproblems3Themethodconsistsinnding(qN;uN)2MNVNsuchthatZKqNrdx=ZKuNrrdx+Z@KbuNrnds;(2.7)ZKqNrvdx=ZKfvdx+Z@KvbqNnds;(2.8)foralltestfunctions(r;v)2S(K)dS(K),forallelementsK2T.ThefunctionsbuNandbqNin(2.5)and(2.6)areth