MULTIGRID IN H(div) AND H(curl)(2)

Abstract. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl) in three dimensions. We show that if appropriate finite el

2DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

ThespacesH(div)andH(curl)arisenaturallyinmanyproblemsof uidme-chanics,solidmechanics,andelectromagnetism.Frequentlytheseapplicationsre-quireafastsolutionmethodforoneorbothoftheequations(1.1).Insomeappli-cations,essentialboundaryconditionsareimposed.Thatis,thebilinearformΛdis (div),thesubspaceofH(div)consistingofvector eldswhosenor-restrictedtoH (curl),malcomponentvanisheson ,orthebilinearformΛcisrestrictedtoH

thesubspaceofH(curl)consistingofvector eldswhosetangentialcomponentvanisheson .Althoughwewillnottreatthissituationexplicitlyhere,there-sultsandanalysiswegiveadapttothecaseofessentialboundaryconditionswithonlyminorandstraightforwardmodi cations.

In§7of[2],wediscussindetailtheapplicationoffastsolversfortheequationΛdhu=ftobothmixedandleastsquaresformulationsofsecondorderellipticboundaryvalueproblems,includingoneinwhichκ 1.Severalotherapplicationsarediscussedbrie yaswell.ApplicationsoffastsolversforΛchp=gariseinvariouscontextsinelectromagnetism.Forexample,insimpletime-discretizationsofMaxwell’sequations,thisoccurswithκproportionaltothetimestep.See[11]foradetaileddiscussion.SuchsolversalsohaveapplicationstosomeformulationsoftheNavier–Stokesequationsasdiscussedin[7]and[8].

Multigridmethodshavebeenestablishedasamongthemoste cientsolversfordiscretizedellipticproblemsandaconsiderabletheoryhasbeendevelopedtojustifytheiruse.See,e.g.,[4],[9],[15].Unfortunately,someofthesimplestandmostfrequentlyusedsmoothersforellipticproblemsdonotyielde ectivemulti-griditerationswhenappliedtotheproblemsconsideredhere(see,forexample,[6]).Thisfailurecanbetracedtoakeydi erencebetweentheoperatorsΛdandΛly,theeigenspaceassoci-atedtotheleasteigenvalueoftheformeroperatorscontainsmanyeigenfunctionswhichcannotberepresentedwellonacoarsemesh(whileloweigenvalueeigenfunc-tionsforstandardellipticoperatorsarealwaysslowlyvarying).ThisisbecausetheoperatorΛdreducestotheidentitywhenappliedtosolenoidalvector elds,althoughitbehaveslikeasecondorderellipticoperatorwhenappliedtoirrota-tionalvector elds.ExactlythereverseholdsforΛc.ItisthereforenotsurprisingthattheHelmholtzdecompositionofanarbitraryvector eldintoirrotationalandsolenoidalcomponentsplaysanimportantroleintheunderstandingandanalysisoftheseproblems.Inparticular,wemakesubstantialuseofdiscreteversionsoftheHelmholtzdecompositioninouranalysisofmultigridmethods.

ThemainresultofthispaperisaproofthatthestandardV-cyclemultigridal-gorithmisane ectivesolverorpreconditionerforproblemsinvolvingtheoperatorscΛdhorΛhinthreedimensionsif(1)appropriate niteelementsubspacesofH(div)andH(curl)aretaken,and(2)appropriatesmoothersareused.Moreprecisely,weshowthatifwetakeVhtobetheRaviart–Thomas–Nedelecspaceofanyorderwithrespecttoatetrahedralmeshofsizeh,andifΘdhistheapproximateinverseofΛdhde nedbytheV-cyclealgorithmusinganyofseveraladditiveormultiplica-

dtiveSchwarzsmoothers,thenI ΘdhΛhisapositivede nitecontraction,whose

normisboundedawayfrom1uniformlyinthemeshsizeh,thenumberofmeshlevels,andtheparametersρ,κ∈(0,∞).Ofcourse,thisimpliesthatΘdhisagood

dpreconditioneraswell:theconditionnumberofΘdhΛhisboundedindependently