Error analysis of a C0 discontinuous Galerkin meth

ErroranalysisofaC0discontinuousGalerkinmethodforKirchho®plates¤XuehaiHuanga,JianguoHuangb;cyaCollegeofMathematicsandInformationScience,WenzhouUniversity,Wenzhou325035,ChinabDepartmentofMathematics,andMOE-LSC,ShanghaiJiaoTongUniversity,Shanghai200240,ChinacDivisionofComputationalScience,E-InstituteofShanghaiUniversities,Shanghai200235,ChinaAbstractThispaperdiscussesapriorierroranalysisofaC0discontinuousGalerkinmethod(CDG)forKirchho®plates(cf.[17]).WiththehelpoferrorestimatesofaL2orthogonalprojectionoperatorandaninterpolationoperatordueto[1,13],weobtaintheerrorestimatefortheCDGmethodinenergynorm,followingthetechniquein[6].TheerrorestimateinH1normisalsoo®eredviathedualityargument.Severalnumericalresultsareperformedtosupportthetheoryobtained.Keywords.Kirchho®plates;C0discontinuousGalerkinmethod(CDG);interpolationoperator;erroranalysis1IntroductionDiscontinuousGalerkin(DG)methodshavebeendevelopedthoroughlyforsolvingbihar-monicequationsandKirchho®platebendingproblems,withtheinteriorpenalty(IP)methodasatypicalone(cf.[2,3,5,12,18{20,22]).OnemainadvantageofDGmethodsforfourthorderproblemsisthatitrequireslessregularityof¯niteelementspacesbyincludingsomeedgetermsintothediscretevariationalformulation.In[12],aC0IPformulationwasdevisedforKirchho®platesandquasi-optimalerrorestimateswereobtainedforsmoothfunctions.In[5],arigorouserroranalysisforthepreviousmethodwasderivedundertheweakregularityassumptionforthesolution[11,15],andapost-processingprocedurewasalsoproducedthatcangenerateC1approximatesolutionsfromtheobtainedC0approximatesolutions.Adrawbackoftheforgoingmethodisthepresenceofadimensionlesspenaltyparameterwhichmustbechosensuitablylargetoguaranteestability,butitcannotbepreciselyquanti¯edapriori.Basedonthisob-servation,anewC0DG(CDG)methodwasdesignedin[23]forwhichthestabilityconditioncanbepreciselyquanti¯ed.ThefullydiscontinuousIPmethodwasinvestigatedsystemati-callyin[18{20,22]forbiharmonicproblems,wherethesubdivisionmeshsizeandthedegreeofpolynomialsateachindividualelementcanvaryarbitrarily,verysuitableforthedesignofhp-adaptivealgorithms.However,duetothefactthatnodalvariablesontheedgesofelementsmaytakedi®erentvalues,thetotalnumberofdegreesoffreedomismuchmorethanthatofusual¯niteelementmethods.In[17],bytakingnumericaltracesintermsofadiscretestabilityidentity,aclassofstableCDGmethodswereproposedforKirchho®platebendingproblems,andtheresultinglocalCDG(LCDG)methodfromtheCDGmethoddoesnotcontainanyto-be-determinedparametersandismoreconvenienttoimplementinactualcomputationthanthatin[23].BasedonCiarlet-Raviartmethod,amixeddiscontinuousGalerkinmethodforbiharmonicequationwasdesignedin[16]usingtheideaofinteriorpenaltymethod.Sincetheglobalmassmatrixofthismethodisblockdiagonal,thecorrespondinglinearsystemcanbereducedtoasmallerlinearsystemconsistingonlyoneunknownapproximationuhofu,justastheLCDGmethod.Byrewritingbiharmonicequationasa¯rst-ordersystemandusingsingle¤Theworkofthe¯rstauthorwaspartlysupportedbytheNNSFC(Grantnos.11126226,11171257)andZhejiangProvincialNaturalScienceFoundationofChina(Y6110240,LY12A01015).TheworkofthesecondauthorwaspartlysupportedbytheNNSFC(Grantnos.11171219,11161130004)andE{InstitutesofShanghaiMunicipalEducationCommission(E03004).yCorrespondingauthor.E-mailaddress:[e-mail]APPLICATIONS,VOL.15,NO.1,118-132,2013,COPYRIGHT2013EUDOXUSPRESS,LLCface-hybridizabletechniquein[9],Cockburn,DongandGuzm¶anderivedahybridizableandsuperconvergentDGmethodin[8]whichimprovestheconvergencerateoftheapproximationto¢uwithorderO(hk+1=2).ThemaingoalofthispaperistoestablishapriorierroranalysisoftheCDGmethodin[17].Itismentionedthattheerroranalysiswasjustobtainedin[17]fortheLCDGmethod.WiththehelpoferrorestimatesofaL2projectionoperatorandaninterpolationoperatordueto[1,13],weobtaintheerrorestimateforgeneralCDGmethodinenergynorm,followingthetechniquein[6].Inaddition,wealsogettheerrorestimateinH1normbyusingdualityargument.Finally,weo®ersomenumericalresultstoillustratetheaccuracyofourCDGmethod,todemonstrateourtheoreticalresults.Therestofthispaperisorganizedasfollows.Somebasicde¯nitionsandsymbolsaregiveninSection2,wheretheCDGmethodforKirchho®platesisalsointroduced.WeobtaintheapriorerrorestimatesinSection3.Inthe¯nalsection,weprovideseveralnumericalresultstodemonstrateourtheoreticalresults.2TheCDGmethodforKirchho®platesLet­½R2beaboundedpolygonaldomainandf2L2(­).ThemathematicalmodelofaclampedKirchho®plateunderaverticalloadf2L2(­)reads[14,21]½r¢(r¢M(u))+f=0in­;u=@nu=0on@­;(2.1)wherenistheunitoutwardnormalto@­,ristheusualgradientoperatoractingontensor¯elds(cf.[21]),andM(u):=(1¡º)K(u)+ºtr(K(u))I;K(u):=(Kij(u))2£2;Kij(u):=¡@iju;1·i;j·2;withIasecondorderidentitytensor,trthetraceoperatoractingonsecondordertensors,andº2(0;0:5).Introduceanauxiliarysecondordertensor¯eld¾by¾:=(1¡º)K(u)+ºtr(K(u))I:Then,Problem(2.1)canbereformulatedasthefollowingsecond-ordersystem:8:11¡º¾¡º1¡º2(tr¾)I=K(u)in­;r¢(r¢¾)=¡fin­;u=@nu=0on@­:(2.2)ACDGmethodbasedon(2.2)isintroducedin[17]forsolvingproblem(2.1).Forrecallingthemethodandlaterrequ