Some new families of finite elements for the stokes equations 1990 Stenberg NM

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1、Numer.Math.56,827-838(1990)NumerischeMathemaUk9Springer-Verlag1990SomeNewFamiliesofFiniteElementsfortheStokesEquationsRolfStenbergINRIARocquencourt,B.P.105,F-78!53LeChesnay,FranceandFacultyofMechanicalEngineering,HelsinkxUniversityofTechnology,SF-02150Espoo,Finl

2、andSummary.WeintroduceawayofusingthemixedfiniteelementfamiliesofRaviart,ThomasandNedelec[13,14],andBrezzietal.[5-7],forconstruct-ingstableandoptimallyconvergentdiscretizationsfortheStokesproblem.SubjectClassifications:AMS(MOS):65N30;CR:G1.8.1IntroductionRecentlyt

3、herehasbeenintroducedanumberofnewmixedfiniteelementspacesfortheapproximationofsecondorderellipticproblems[5-7].Thesenewmeth-odsareofthesametypeasthemethodsintheclassicalRaviart-Thomas-Nedelecfamilies[13,143,buttheydifferinthattheyaremoreaccurateforthesamecomputat

4、ionalcost.Oneofthereasonsfortheincreasedinterestinthisfieldisthesuccessofthesemethodsincertainapplications.Forsomeproblemsingeophysicsandsemiconductorphysicstheyhavebeenshowntobemoreefficientthanmoreconventional"displacementmethods";cf.[8,11,19]andthereferencesth

5、erein.Inthisnotewewillconsidertheapplicationofthesespacesfortheapproxi-mationoftheStokesequations:Findthevelocitynandthepressurepsuchthat-Au+Vp=finK2,(1.1)divu=OinO,U=Uoona~2,whereforsimplicityf2cRN,N=2,3,isassumedtobeaboundedpolygonalorpolyhedraldomain.Letusassu

6、methatfeL2(f2)NanduoeH1/2(~Q)Nwithuo.nds=O828R.Stenbergsothatthereisauniquesolutionto(1.1)(asusualthepressureisnormalizedtohaveazeromeanvalueoverf2).WhenderivingoptimalL2-estimatesforthevelocitywewillneedtheusualregularityassumptionforthesolutionto(1.1)whenno=0(1

7、.2)Ilull=+IlpEI1~Cltflfo.Belowwewillshowthatthespacesof[5-7,13,14-]canbeusedasbuildingblocksforproducing"Taylor-Hood"typemethodsfortheStokesproblem,i.e.methodswithacontinuousapproximationforthepressure.Letusherealsomentionthatwein[16]usedthesamebasicfiniteelement

8、spacesfortheconstructionofafamilyforthediscretizationoftheequationsoflinearelasticity.Theestimatesof[16]areuniformlyvalidwithrespecttothePoissonratio,andhencea