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1、AMIXEDFINITEELEMENT~,~THODFOR2~ndORDERELLIPTICPROBLEMSP.A.RaviartandJ.M°ThomasI.INTRODUCTIONLet~beaboundedopensubsetofRnwithaLipshitzcontinuousboundaryF.Weconsiderthe2ndorderellipticmodelproblem-Au=fin~,(I.I)Iu=0ohF,wherefisagivenfunctionofthespaceL2(~).Avar
2、iationalformofproblem(1.1),knownasthecomplementaryenergyprinciple,consistsinfindingp=graduwhichminimizesthecomplementaryenergyfuncti2nal(1.2)~(q)~=yI~I~dxnovertheaffinemanifoldWofvector-valuedfunctions~@(L2(~))n%whichsatisfytheequilibriumequation(1.3)div~+f=
3、0in~.TheuseofcomplementaryenergyprincipleforconstructingfiniteelementdiscretizationsofellipticproblemshasbeenfirstadvocatedbyFraeijsdeVeubeke[51,[6],
4、7].Theso-calledequilibriummethodconsistsfirstinconstructingafinite-dimensionalsubmanifold~hofW~andtheninfind
5、ing~h6~hwhichminimizesthecomplementaryenergyfunctionalI(q)overtheaffinemanifold%~h.For2ndorderellipticproblems,thenumericalanalysisoftheequilibriummethodhasbeenCentredeMath~matiquesAppliqu~es,EcolePolytechniqueandUnive~sit~deParisVI.~Universit~deParisVI.293m
6、adebyThomas[19],[20].Now,wenotethatthepracticalconstructionofthesubmanifold~hisnotingeneralasimpleptoblemsinceitrequiresasearchforexplicitsolutionsoftheequilibriumequation(1.3)inthewholedomain~.Inordertoavoidtheabovedifficulty,wecanuseamoregeneralvariational
7、principle,knowninelasticitytheoryastheHellinger-Reissnerprinciple,inwhichtheconstraint(1.3)hasbeenremovedattheexpensehoweverofintroducingaLagrangemultiplier.Thispaperwillbedevotedtothestudyofafiniteelementmethodbasedonthisvariationalprinciple.Infact,thisso-c
8、alledmixedmethodhasbeenfoundveryusefulinsomepracticalproblemsandreferto[17]foranapplicationtothenumericalsolutionofanonlinearproblemofradi~tivetransfer.Forsomegeneralresultsconcerningmixedmethods,werefertoOden[12],[13],Oden&Reddy[14],Reddy[16].Mixedmethodsfo
9、rsolving4thorderellipticequationshavebeenparticularlyanalyzed:seeBrezzi&Raviart[2],Ciarlet&Raviart[4],Johnson[9],[I0],andMiyoshi[11].ForrelatedresultswereferalsotoHaslinger&Hl~vacek[8].Anoutline
