Adaptive least squares finite element methods for

ADAPTIVELEASTSQUARESFINITEELEMENTMETHODSFORTHESTOKESPROBLEMIng-JerLinandDa-PanChenDepartmentofMechanicalEngineering,NationalChiaoTungUniversity,Hsinchu,Taiwan,R.O.C.Jinn-LiangLiuDepartmentofAppliedMathematics,NationalChiaoTungUniversity,Hsinchu,Taiwan,R.O.C.November12,1996Keywords:Adaptive,Stokesproblem,Least-squaresniteelementmethod,ErrorestimationABSTRACTAdaptiveleast-squaresniteelementmethods,includingthestandardandtheweightedversions,fortheStokesprobleminthevelocity-vorticity-pressureformulationarepresentedinthearticle.Themostsignicantfeaturesoftheproposedadaptivemethodsarethattheaposteriorierrorestimatorsdonotinvolveuxjumpsacrossinterelementboundaries,thatthelocalproblemsforerrorestimationdonotrequiretheLadyzhenskaya-Babuska-Brezziconditiontobesatised,andthatnoboundaryconditionsarerequiredforcalculatingtheerrorsonanelement-by-elementbasis.Moreover,forboththestandardandtheweightedleast-squaresmethods,theerrorestimationproceduresarealmostidentical.Conse-quently,theadaptivemethodscanbeeasilyincorporatedintotheexistingnonadaptivecodes.Numericalexperimentsareprovidedtoillustratethequalityandreliabilityoftheproposedmethods.INTRODUCTIONRecently,therehasbeensubstantialinterestintheuseofleast-squaresprinciplefortheapproximatesolutionoftheStokesprobleminvelocity-vorticity-pressure(VVP)formulation(BochevandGunzburger,1993;1994;ChangandJiang,1990;Chang,1994;JiangandChang,1990).Theleast-squaresapproachoerscertainadvantageousfea-turescomparedtotheclassicalGalerkinmixedformulation.Forexample,thechoiceofniteelementspacesintheleast-squaresformulationisnotsubjecttotheLBBconditionandasinglecontinuouspiecewisepolynomialspacecanbeusedfortheapproximationofallunknowns.Theapproachyieldssymmetric,positivedenitelinearsystemwhichcanbeeectivelysolvedby,e.g.,conjugategradientmethods.Itdoesnotincurarti-cialconditionsforthevorticityontheboundarywherethevelocityisspecied.Finally,accurateapproximationscanbeobtainedforallvariables,includingthevorticity.Inengineeringapplications,thediscretizationerrorisinevitablewhenanumericalmethodisusedtoapproximatethesolutions.Althoughengineerscanevaluatetheaccuracyoftheniteelementsolutionwiththeirexperiences,itisstilldesirabletoadaptthesolutionproceduresinordertoecientlyestimatetheerrorandeectivelycontroltheerrorandconsequentlytooptimizethecomputingresources.Therapidlygrowingadaptivemethodology(Demkowiczetal.,1989;Odenetal.,1989;1990;Peraireetal.,1992;Rachowiczetal.,1989;Zienkiewiczetal.,1989;ZienkiewiczandZhu,1990;ZhuandZienkiewicz,1990)isaimingtoaidtheseneedsinpractice.Adaptivemethodsinvolvetwobasicprocesses:aposteriorierrorestimationandmeshrenement.Theformercanberegardedasthekerneloftheadaptivescheme.Ouradaptiveschemeforthestandardandtheweightedleastsquaresmethods(SLSFEMandWLSFEM)isbasedontheweakresidualaposteriorierrorestimationproposedin(Liu,1996)inwhichageneralframeworkoftheestimationisdevelopedinanabstract1variationalsetting.Wesummarizethemainadvantagesofourapproachwhencomparedwiththepreviouserrorestimation.First,theerrorestimationdoesnotinvolvetheuxjumpsontheinterfacesofelements.ThisisoneofthemajorconcernsinothererrorestimatorsfortheStokesequations(BankandWelfert,1990;Verfurth,1989).Ourapproachusesinsteadaproperconstructionoflocalshapefunctionsforerrorestimationwhichisbasedonaformulacompletelysimilartothatofapproximation.Thismeansthatthesamecodeusedforapproximationcanalsobeusedforerrorestimation.Hence,asimpleimplementationoftheerrorestimatorcanbeexpected.Wereferto(Liuetal.,1996)formoredetailsrelatedtotheimplementationissues.Second,ourerrorestimatorinherentlyavoidsthevericationoftheconsistencyLBBconditionfortheestimationsubspacesduetothenatureofLSFEM.Otherwise,somestabilizationtermsmustbeintroducedforlocalerrorestimationwhenthemixedmethodisusedfortheStokesproblem;see,e.g.,(BankandWelfert,1990).Thestabilizationtermsaresomewhatadhoc.Third,ourapproachfurthersimpliestheimplementationwithoutrequiringanyboundaryconditionsbeprescribedforthelocalproblems.ItthusprovidesauniederrorestimationforbothSLSFEMandWLSFEM.Toourknowledge,itseemstobetherstuniedapproachforbothadaptiveSLSFEMandWLSFEM.Theremainderofthepaperisarrangedasfollows.InSection2,thevelocity-vorticity-pressureformulationoftheStokesequationsisbrieydescribed.TheSLSFEMandWLSFEMfortheVVPformulationoftheStokesproblemarethengiveninSection3.TheaposteriorierrorestimatorispresentedinSection4.WealsoincludeanadaptivealgorithminSection5.NumericalresultsshowingtheeectivenessoftheresultingadaptivemethodsandthequalityoftheerrorestimatorareillustratedinSection6.WemakesomeconcludingremarksinSection7.THEVELOCITY-VORTICITY-PRESSUREFORMULATIONOFSTOKESEQUATIONSLetR2beaboundeddomainwithsmoothboundary@.The2DStokesproblemforincompressibleowisgivenby(u+gradp=fin;divu=0in;(1)whereistheinverseofReynoldsnumber,u=(u1;u2)Tthevelocity,pthetotalheadofpressure,andf=(f1;f2)Tthegivenbodyforce.Hereallvariablesarenondimensionalizedbyacharacteristicvelocityandlengthscale,andbythedensityofuid.Introducingthevorticity!=curluasanauxiliaryvariable(Ch