mixed finite element in 3d in hdiv and Hcurl Nedelec 1986

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1、MIXEDFINITEELEMENTIN3DINH(div)ANDH(curl)J.C,NEDELECEcolePolytechnique,CentredeMathdmatiquesAppliqudes91128Palai~eau,FranceI.INTRODUCTION.FrayesDeVenbekefirstintroducethemixedfiniteelement.ThenP.A.RaviartandJ,M.Thomasdoessomemathematicsontheseelementin2Dandothersdoalso:F.Br

2、ezziV.Babuska...In1980weintroduceafamilyofsomemixedfiniteelementin3DandweusethemforsolvingNavierStokesequations.In1984F,Brezzi,J.DouglassandL.D.Mariniintroducein2DanewfamilyofmixedfiniteelementconforminginH(div).ThatpaperwasthestartingpointforbuildingnewfamiliesOffiniteele

3、mentin3D,II.FINITEELEMENTINH(div).Notations.Kisatetrahedron~KitsboundarynthenormalfafacewhichareaisId2fraisanedgewhichlenghtisJ~dscurlu=V^uu=(ul,u2,u3)H(curl)={uEL2(~))3;curluE(L2(~))3}div=V.uH(div)={uE(L2(~))3;divu@L2(~)}Spacesofpolynomials.Pk=polynomialsofdegreelessorequ

4、altok~k="homogeneousofdegreekx!Dk=(Pk_l)3+PNk_lrr=x2Ix3Sk={pE(pk);(r.p)~0}=(Pk_

5、)3~Sk322dimSk=k(k+2)dim~k=(k+3)(k2+I)k=(k+3)(k+2)kdimk2WearenowabletointroducethefiniteelementconforminginH(div).Definition.WedefinethefiniteelementbyI)Kisatetrahedron2)P=(Pk)3isaspaceofpolynom

6、ials3)Thesetofdegreesoffreedomwhichare(3.1)(p.n)qdy;VqEpk(f);f(3.2)jf(p.q)dx;VqE~k-1KwehavetheTheorem.TheabovefiniteelementisunisolventandconforminginH(div).Theassociatein-terpolationoperatorHissuchthatdiv~p=9"divp;VpEH(div),where~*istheL2projectiononPk-1"Whenk=I,thecorres

7、pondingelementhasnointeriormomentsand12degreesoffree-dom.Itsdivergenceisconstant.Proposition.Foratetrahedron"regularenough"whichdiameterisk,wehaveIIp-~pN(L2(K))3

8、initeelementissaidtobeconforminginafunctionalspaceiftheinterpolateofanelementofthisspacebelongtothisspace.Inourcase,theconformityinH(div)isequivalenttothecontinuityofthenormalcomposentateachinterface.Thispropertyisclearlytrueforourfiniteele-mentsincetheunknownsonthefaceare

9、I(p.n)qdy;VqEPk(f)fandp.nisalsoPk(f).323III.FINITEELEMENTINH(curl).Afiniteelementisconfor