Mixed Finitc Elements in R3 Nedelec 1980

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1、Numer.Math.35,315-341(1980)NumerischeMathemaUk9bySpringer-Verlag1980MixedFinitcElementsinIR3J.C.NedelecCentredeMath6matiquesAppliqu+es-EcolePolytechnique,91128PalaiseauCedex,FranceSummary.WepresentheresomenewfamiliesofnonconformingfiniteelementsinIR3.Thesetwofamiliesoffini

2、teelements,builtontetrahedronsoroncubesarerespectivelyconforminginthespacesH(curl)andH(div).WegivesomeapplicationsoftheseelementsfortheapproximationofMaxwell'sequationsandequationsofelasticity.SubjectClassifications:AMS(MOS):65N30,CR:5.17.First,weintroducesomenotations:Kisa

3、tetrahedronoracube,thevolumeofwhichis.[dx;K~Kisitsboundary;fisafaceofK,thesurfaceofwhichis~dy;faisanedge,thelengthofwhichis~ds;LZ(K)istheusualHilbertspaceofsquareintegrablefunctionsdefinedonK;Hm(K)={qSELZ(K);O'O~L2(K);tc~l

4、=VAu,(definedbyusingthedistributionalderivative)forU=(Ul,U2,U3);uieLZ(K);H(curl)={u~(Lz(K))3;curlue(U(K))3};divu=F-u;H(div)={ue(U(K))3;divueL2(K)};Dkuisthek-thdifferentialoperatorassociatedtou,whichisa(k+1)-multilinearoperatoractingon]R3;kisanindex;isthelinearspaceofpolynom

5、ials,thedegreeofwhichislessorequaltok;isthegroupofallpermutationsoftheset{1,2....,k};Corc~willstandforanyconstantdependingpossiblyon~.1.StudyoftheFiniteElementsBuiltonTetrahedrons1.1SomeSpaceofPolynomialsonIR"Weintroduceheresomelinearspacesofpolynomialswhichwillbeusedlatert

6、obuildtheconformingfiniteelementsinH(curl)orinH(div).WeconsiderherethecaseofIR"thoughhereafterwewillonlyusen=2andn=3.0029-599X/80/0035/0315/$05.40316J.C.NedelecDefinition1.Forue(Ck(IR"))",ekuisthek+1multi-linearformobtainedfromDRubythefollowingsymmetrization:V~1,~2.....~k,~

7、k+1elR",1eku(~l,~2.....~k,~k+l)=~,(k+l)!Dku(~oo).....~(k),~(k+l))"(1)~rEak+1Remark.Whenk=1,euisthefollowingbilinearsymmetricform:1[OuiOujeuisthestraintensorofelasticitywhenn=3.9Letusproveapropertyoftheoperatoreku.Letet,...,e,bethebasisinIR".WehaveLemma1.Forc~=(cq.....~,),

8、amulti-index,]c~]=k+1,wehave0k+r~~t-times~zz-l-timesaj+i-timescry-times]luJ--[~iT-