MULTIGRID IN H(div) AND H(curl)(3)

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

MULTIGRIDINH(div)ANDH(curl)3

ofh,thenumberofmeshlevels,andρandκ.Tode netheSchwarzsmoothers,wecanuseadecompositionofVhintolocalpatchesconsistingofallelementssur-roundingeitheranedgeoravertex,orathirddecompositioncanbeusedbasedontheHelmholtzdecomposition(see(4.2)).PreciselyanalogousresultsholdintheH(curl)caseifwetakeQhtobetheNedelecedgespacesofanyorder.Inthiscase,thesmootherscanbebasedeitheronadecompositionbasedonvertexpatchesoronthedecomposition(4.4)arisingfromtheHelmholtzdecomposition.

TheresultsofthispapergeneralizetothreedimensionsoneswhichweobtainedforH(div)intwodimensions[2].ThespacesH(div)andH(curl)areessen-tiallythesameintwodimensions,andsoouranalysisofmultigridin[2]adaptstoH(curl)withonlythemostmechanicalchanges.Inthreedimensions,whiletherearemanysimilaritiesbetweenH(div)andH(curl),therearealsosigni cantdi erences,especiallybetweentheir niteelementdiscretizations.Forthisreason,theanalysisforH(curl)requiresanumberofadditionalideas.Inourpresentation,wehavestressedthesimilaritybetweentheH(div)andH(curl)casesasmuchaspossible,limitingthedi erencestotheproofsofthetwo-levelestimatesformixedmethodsinthe nalsection,wheretheyaredescribedexplicitly.

The rstresultsformultigridinH(div)inthreedimensionsareduetoHiptmairin[10].Thesameauthorobtainedthe rstresultsformultigridinH(curl)in[11].Auni edandsimpli edtreatmentofthoseimportantworksisgivenbyHiptmairandToselliin[12].Ourresultsarecloselyrelatedtotheresultsin[10],[11],[12],andsomeofourargumentsderivefromthem.Themajordi erencebetweenourapproachandtheirsisthatweemployamultigridframeworkaspresented,forexample,in[4],andverifythehypothesesrequiredbythisapproachbydevelopingnecessaryestimatesformixed niteelementmethodsbasedondiscretizationsofH(div)andH(curl).Speci cally,wetakeTheorem3.1belowasthebasisforouranalysis,anddeveloptwo-levelestimatesformixedmethodsin§5inordertoapplythistheorem.Bycontrast,HiptmairandToselliuseanoverlappingSchwarzmethodframeworkaspresented,forexamplein[13].Animportantbene tofourapproach,whichisalsosomewhatlesscomplicated,isthatweobtainestimateswhichareindependentoftheparametersρandκoccuringinthebilinearform.Bycontrast,in[11],theconditionnumberofΘhΛhisonlyshowntobeO(1/κ3)whenρ=1andκissmall(andthecaseofκ/ρlargeisnotdiscussed).

Concerningnotation,weuseboldfacetypeforvector-valuedfunctions,operatorswhosevaluesarevector-valuedfunctions,andspacesofvector-valuedfunctions.ThenormintheSobolevspacesHs( )andHs( )arebothdenotedby · s,withtheindexs=0suppressed.ThenormassociatedtothebilinearformΛdisdenoted · Λd,orsimply · H(div)ifρ=κ=1,andanalogouslyforthenormassociatedtoΛc.

Weconcludewithanoutlineoftheremainderofthepaper.Inthenextsec-tion,weintroducethe nitedimensionalsubspacesofH(div)andH(curl)thatweshallconsiderinthispaper,namelytheRaviart–Thomas–NedelecspacesandNedelecedgespaces,respectively.Wethenstatesomeofthekeypropertiesofthesespaceswhichweshalluseinthesubsequentanalysis,themostimportantofwhicharediscreteHelmholtzdecompositionsofeachspace.In§3,westatesomestandardresultsformultigriditerations,inordertoisolatesu cientconditionsonadditiveandmultiplicativeSchwarzsmoothersfore cientalgorithms.In§4,we