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1、ANALYSISOFTHEFINITEELEMENTMETHODFORTRANSMISSION/MIXEDBOUNDARYVALUEPROBLEMSONGENERALPOLYGONALDOMAINSHENGGUANGLI,ANNAMAZZUCATO,ANDVICTORNISTORAbstract.Westudytheoreticalandpracticalissuesarisingintheimple-mentationoftheFiniteElementMethodforastronglyellipticsecondorderequationwithjumpdisconti
2、nuitiesinitscoefficientsonapolygonaldomainΩthatmayhavecracksorverticesthattouchtheboundary.Weconsiderinparticulartheequation−div(A∇u)=f∈Hm−1(Ω)withmixedboundaryconditions,wherethematrixAhasvariable,piecewisesmoothcoefficients.WeestablishregularityandFredholmresultsand,undersomeadditionalcon-dit
3、ions,wealsoestablishwell-posednessinweightedSobolevspaces.WhenNeumannboundaryconditionsareimposedonadjacentsidesofthepolygonaldomain,weobtainthedecompositionu=ureg+σ,intoafunctionuregwithbetterdecayattheverticesandafunctionσthatislocallyconstantnearthevertices,thusprovingwell-posednessinana
4、ugmentedspace.Thetheoreticalanalysisyieldsinterpolationestimatesthatarethenusedtocontructimprovedgradedmeshesrecoveringthe(quasi-)optimalrateofconvergenceforpiecewisepolynomialsofdegreem≥1.Severalnumericaltestsareincluded.IntroductionInthispaperwestudysecond-order,stronglyellipticoperatorsi
5、ndivergenceformP=−divA∇ongeneralizedpolygonaldomainsintheplane,wherethecoefficientsarepiecewisesmoothwithpossiblyjumpdiscontinuitiesacrossafinitenumberofcurves,collectivelycalledtheinterface.LetΩbeaboundedpolygonaldomainthatmayhavecurvedboundaries,cracks,orverticestouchingtheboundary.Wereferto
6、suchdomainsasdomainswithpolygonalstructure(seeFigure1foratypicalexample).WeassumethatΩ=¯∪Ω¯j,whereΩjaredisjointdomaiswithapolygonalstructuresuchthattheinterfaceΓ:=∪∂Ωjr∂Ωisaunionofdisjoint,piecewisesmoothcurvesΓk.ThecurvesΓkareallowedtointersecttransversely.Weareinterestedinthenon-homogeneo
7、ustransmission/mixedboundaryvalueproblemPu=finΩ,DPu=gon∂Ω,u=gon∂Ω,νNNDD(1)u=uandDP+u=DP−uonΓ,+−ννandtheconvergencepropertiesofitsFiniteElementdiscretizations.Here,A=P(A)isthesymmetricmatrixofcoefficientsofP,DP:=νiA∂istheconor-ijνijijjmalderivativeassociate
