MULTIGRID IN H(div) AND H(curl)(9)

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)9

4.MultigridconvergenceinH(div)andH(curl).Weconsideranestedsequenceofquasi-uniformtetrahedralmeshesTj,1≤j≤J.Thesegiveriseto

cspacesWj,Qj,Vj,andSjandoperatorsΛdj:Vj→VjandΛJ:Qj→Qj.In

thissection,weuseTheorem3.2toobtainaconvergenceresultforthemultigridcV-cycleappliedtotheequationΛdJu=forΛJp=ginthespaceX=VJorQJ.FortheenclosingHilbertspaceYwetakeL2.Wenotethatproperties(3.1)and(3.2)onlyinvolvesubspacesattwolevels.LethdenotethemeshsizeofsomemeshTjandletHdenotethemeshsizeofthenextcoarsermeshTj 1.Tosimplifynotation,weshallwriteThandTHforTjandTj 1,andsimilarlyinothercaseswherethesubscriptsjandj 1arise.

Tode netheSchwarzsmoothers,wemustdecomposethespaceVhorQh.ForthespaceVh,threepossibledecompositions,basedonfacepatches,edgepatches,andvertexpatches,aregivenin(2.1).FromthepointofviewofimplementationofthecorrespondingSchwarzsmoother,theface-baseddecomposition,whichhasonlytwoelementsperpatch,ismoste cient,theedge-basedlesse cient,andthevertex-basedSchwarzsmoothertheleaste cient.However,asourtheorywillsuggestandnumericalcomputationsinanalogoussituationsreinforce[6],theface-basedSchwarzsmootherdoesnotleadtoane cientmultigridalgorithm.Belowweshallprovethatbothdecompositions ev(4.1)Vh=VhandVh=Vh,

e∈Ehv∈Vh

Theimplementationofthecorrespondingsmoother,whichmaybemoree cientthanthesmootherbasedonedgepatches,isdiscussedin[hiptmair-hdiv].Ouranalysisbelowappliestothissmootheraswell.

ForthespaceQhwemayuseeitherthedecomposition (4.3)Qh=Qvh,

v∈VhyieldSchwarzsmoothersthatsatisfytheconditionsofTheorem3.2withconstantsindependentofhandκ.In[10]Hiptmairgeneralizestothreedimensionadecom-positionusedintwodimensionsbyVassilevskiandWang[14],namely, f (4.2)Vh=Vh+curlQeh.f∈Fhe∈Eh

oroneduetoHiptmair[11],

(4.4)Qh=

e∈Eh

Itiseasytocheckthatsincenopointbelongstomorethansixofthe ehor

ffourofthe vhor h,allthesedecompositionssatisfythecondition(3.1)withβ

independentofh,ρ,andκ(βwillneverexceed10).Itthusonlyremainstoverifycondition(3.2),whichwestatefortheparticularcaseofthe rstsmootherin(4.1)andthesmootherin(4.3)inthefollowingtwotheorems.Theveri cationfortheothersmootherswillberemarkedonbelow.Forthesetheorems(only)werequiretheboundedre nementhypothesisH≤ch.(Inpractice,valuesofcaround2arecommon.) Qeh+v∈Vh vgradWh.