Finite element methods and their convergence for e

FiniteElementMethodsandTheirConvergenceforEllipticandParabolicInterfaceProblemsZhimingChen1,JunZou2AbstractInthispaper,weconsidertheniteelementmethodsforsolvingsecondorderellipticandparabolicinterfaceproblemsintwo-dimensionalconvexpolygonaldomains.NearlythesameoptimalL2-normandenergy-normerrorestimatesasforregularproblemsareobtainedwhentheinterfacesareofarbitraryshapebutaresmooth,thoughtheregularitiesofthesolutionsarelowonthewholedomain.Theassumptionsontheniteelementtriangulationarereasonableandpractical.MathematicsSubjectClassication(1991):65N30,65F10.Arunningtitle:Finiteelementmethodsforinterfaceproblems.Correspondenceto:Dr.JunZouEmail:[e-mail]:(852)260351541InstituteofMathematics,AcademiaSinica,Beijing100080,P.R.China.Email:[e-mail]rtiallysupportedbyChinaNationalNaturalScienceFoundation.2DepartmentofMathematics,theChineseUniversityofHongKong,Shatin,N.T.,HongKong.E-mail:[e-mail]rtiallysupportedbytheDirectGrantofCUHKandIncentiveFundfromFaculyofScience,CUHK.11IntroductionNumericalsolutionsofsecondorderellipticandparabolicproblemswithdiscontinuousco-ecientsareoftenencounteredinmaterialsciencesanduiddynamics.Itisthecasewhentwodistinctmaterialsoruidswithdierentconductivitiesordensitiesordiusionsareinvolved.Whentheinterfaceissmoothenough,thesolutionoftheinterfaceproblemisalsoverysmoothinindividualregionsoccupiedbymaterialsoruids,buttheglobalregularityisusuallyverylow,seeLittmanetal.[22],Kellogg[15,16],Ladyzenskajaetal.[17].Be-causeofthelowglobalregularityandtheirregulargeometryoftheinterface,achievingthehighorderofaccuracyseemsdicultwithniteelementmethods(cf.Babuska[1]),whoseelementscouldnottwiththeinterfaceofgeneralshape.Babuska[1]studiedtheellipticinterfaceproblemdenedonasmoothdomainwithasmoothinterface.Theinterfaceproblemwasformulatedasanequivalentminimizationprob-lemwithalltheboundaryandjumpconditionsincorporatedinthecostfunctions.Theniteelementmethodswerethenusedtosolvetheminimizationproblems.Undersomeapproxi-mationassumptionsonniteelementspaces,theenergy-normerrorestimateswereobtained.Xu[25]consideredsolvingtheellipticinterfaceproblemassumingitssolutionandthenor-malderivativesofthesolutioncontinuousacrosstheinterface,bythestandardniteelementmethod.Thealgorithmsin[1]and[25]requiretheexactcalculationoflineintegralsontheboundaryofthedomainandontheinterface,andexactintegralsoninterfaceniteelementsarealsoneeded.Han[12]proposedaninniteelementmethod,whichmaybeconsideredasacertainschemeofmeshrenement,forellipticinterfaceproblemswithinterfacesconsistingofstraightlines,notsuitableforcurvedinterfaces.Theenergy-normerrorestimateswereachievedbothin[12]and[25].LeVeque-Li[19]proposedanimmersedinterfacemethodforellipticinterfaceproblemsdenedonaregulardomainforwhichauniformrectangulargridcanbeused.Thennitedierencemethodswereconstructedbasedontheuniformgridandthejumpconditionsontheinterface.Theauthorsappliedtheirmethodsalsoforotherinterfaceproblems,e.g.theStokesowproblem[18],theone-dimensionalmovinginterfaceproblem[20]andHele-Shawow[21].Theresultantlinearsystemsfromthesemethodsarenon-symmetricandindeniteeventheoriginalproblemsareself-adjointanduniformlyelliptic.Theconvergenceproofsofthesemethodsarestillopen.Inthispaper,weproposetheniteelementmethodforsolvingsecondorderbothellipticandparabolicinterfaceproblemsandprovethatthemethodconvergesnearlyinthesameoptimalwayastheusualnon-interfaceellipticandparabolicproblems,bothfortheenergy-normandL2-norm.Infact,theenergy-normerrorestimatecanbeshowntobeoptimal.Theinterfaceisallowedtobeofarbitraryshapebutissmooth.TheresultantlinearsystemsarealwayssymmetricandpositivedenitewhentheoriginalPDEsareself-adjointanduniformlyelliptic.Andinparticular,thedomaindecompositionmethods,whichhavebeeninvestigatedwidelyinrecentyears(cf.Chan-Zou[5]andXu-Zou[26]),canbeappliedheretoconstructecientpreconditionediterativemethodsforsolvingtheselargescaleandsparselinearsystemsofequations.Anddierentfromthepreviousniteelementmethods,thecalculationsofthestinessmatrixandtheinterfaceintegralrelatedtothejumpsofnormalderivativesaremuchsimplerandmorepracticalhere.Consideringweapproximatethesmoothinterfacebyapolygonandtheinterfacefunc-2tionbyitsinterpolant,theapproximationproblemhereseemssimilartotheclassicalniteelementmethodswithstraighttrianglesforsolvingNeumannproblemsforellipticequationsonsmoothdomains(see,forexample,[3,2,8,9]andthereferencestherein).Buttherearesomedierences.Intheclassicalcase,onecanassumefullregularity(orevenmore)aboutsolutions,coecients,boundaryfunctionetc.Butforthecurrentinterfaceproblems,weusu-allyhaveverylowglobalregularityaboutthesolutions,coecientsandinterfacefunctions.Sotheclassicalanalysisisdiculttoapplyfortheconvergenceanalysisintheinterfaceproblem.SomecrucialtechnicaltoolsusedherearesomeSobolevembeddinginequality,Sobolevextensiontheorem,dualarguments,interfaceenergy-normprojectionoperatoranddiscreteL2projectionoperatoretc.Letusnowendthissectionwithsomenotation