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1、MixedFiniteElementMethodsRicardoG.Duran´DepartamentodeMatematica,FacultaddeCienciasExactasyNaturales,Universidadde´BuenosAires,CiudadUniversitariaPabellonI,1428BuenosAires,Argentina´rduran@dm.uba.ar1IntroductionFiniteelementmethodsinwhichtwospacesareusedtoapprox
2、imatetwodifferentvariablesreceivethegeneraldenominationofmixedmethods.Insomecases,thesec-ondvariableisintroducedintheformulationoftheproblembecauseofitsphysicalinterestanditisusuallyrelatedwithsomederivativesoftheoriginalvariable.Thisisthecase,forexample,intheel
3、asticityequations,wherethestresscanbeintroducedtobeapproximatedatthesametimeasthedisplacement.Inothercasestherearetwonaturalindependentvariablesandso,themixedformulationisthenaturalone.ThisisthecaseoftheStokesequations,wherethetwovariablesarethevelocityandthepre
4、ssure.Themathematicalanalysisandapplicationsofmixedfiniteelementmethodshavebeenwidelydevelopedsincetheseventies.AgeneralanalysisforthiskindofmethodswasfirstdevelopedbyBrezzi[13].WealsohavetomentionthepapersbyBabuska[9]andbyCrouzeixandRaviart[22]which,althoughforpa
5、rticularprob-ˇlems,introducedsomeofthefundamentalideasfortheanalysisofmixedmethods.Wealsoreferthereaderto[32,31],wheregeneralresultswereobtained,andtothebooks[17,45,37].Therestofthisworkisorganizedasfollows:inSect.2wereviewsomebasictoolsfortheanalysisoffiniteelem
6、entmethods.Section3dealswiththemixedfor-mulationofsecondorderellipticproblemsandtheirfiniteelementapproximation.WeintroducetheRaviartThomasspaces[44,49,41]andtheirgeneralizationtohigherdimensions,provesomeoftheirbasicproperties,andconstructtheRaviartThomasinterpo
7、lationoperatorwhichisabasictoolfortheanalysisofmixedmeth-ods.Then,weproveoptimalordererrorestimatesandasuperconvergenceresultforthescalarvariable.Wefollowtheideasdevelopedinseveralpapers(seeforexam-ple[24,16]).AlthoughforsimplicityweconsidertheRaviartThomasspace
8、s,theerroranalysisdependsonlyonsomebasicpropertiesofthespacesandtheinterpo-lationoperator,andtherefore,analogousresultsholdforapproximationsobtainedwithotherfiniteelem
