Optimal a priori estimates for interface problems

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1、ptimalaprioriestimatesforinterfa eproblemsi haellumandChristianWienersAbstra t.We onsideraprioriestimatesinweightednormsforinterfa eprob-lemswithpie ewise onstantdiusion onstantswhi hdonotdependontheratiobetweenthe onstants.urresultgeneralizesanestimateofemrabettoarbi-trarydimensionsandin l

2、udes urvedboundaries.Furthermore,wedis uss riteriafortheexisten eofauniformoin areestimateinweightednorms.
ntheaÆrma-tive aseweobtainarobustniteelementerrorboundinweightednorms.Finally,wepresentnumeri alexperimentsin ludinga asewithnouniformoin are on-stant.Thesolutionofanellipti problemwith

3、smooth oeÆ ientsinadomainissmoothintheinterior,andformany asesallboundarysingularities anbe lassied, f.Grisvard[7℄.
fthe oeÆ ientsarepie ewisesmooth,regularitygetslostattheinterfa e;nevertheless,forpie ewise onstant oeÆ ientsasingularity lassi ationispossibleforawiderangeofproblems, f.i aise[1

4、2℄forpolygonaldomains.
nthis ontext,aprioriestimatesforthesmoothpartofthesolutionarerequired.Here,weareinterestedinestimatesforthesmoothpartofthesolutionwhi hinadditionareindependentofthe oeÆ ientsinappropriateweightednorms,i.e.,inrobustestimates,andwegeneralizetheresultstothe aseofnonpolygonaldo

5、mains.Su hrobustestimatesprovidee.g.robustniteelementerrorestimates,whi hinparti ular,togetherwitharobustsmoothingproperty[10,14℄,leadtoparameter-independentmultigrid onvergen e.1We onsiderthefollowingmodelproblem:ndu2H()su hthat01a(u;v)=(f;v)forallv2H();(1)0d2whereRisaboundedips hitzdomain,f

6、2(),andaistheellipti bilinearformZX[a(u;v)=ru
rvdx;=;kkk=1k=1kwithnon-overlappingips hitzsubdomainsand onstant oeÆ ients>0.kkWeassumethat,besidestheips hitzproperty,isapie ewisesmoothboundaryk(inthesensethatsome losedmeasure-zerosubsetZexistssu hthatnZkkkk2isC-smooth;apre iseformulati

7、onoftherequirementsisgivenin )inSe tion1).denotestheouterunitnormalon.nnZwe andenethese ondkkkkfundamentaltensorS(x)(withrespe tto),themean urvatureH(x),andthekkkmaximalprin ipal urvature(x);weassumethatHandare