MULTIGRID IN H(div) AND H(curl)(10)

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

10DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

dTheorem4.1.AssumethatH≤chandthatv∈(I P)Vhbegiven.ThereH eexistsadecompositionv=e∈Ehve,whereve∈Vh,andaconstantγdepending

oncbutindependentofh,ρ,andκsuchthat

e∈Eh

cTheorem4.2.AssumethatH≤chandthatq∈(I P)Qhbegiven.ThereH vvvexistsadecompositionq=v∈Vhq,whereq∈Qh,andaconstantγdepending

oncbutindependentofh,ρ,andκsuchthat Λd(ve,ve)≤γΛd(v,v).

v∈Vh Λc(qv,qv)≤γΛc(q,q).

Toprovetheseresults,wewillmakeuseofthediscreteHelmholtzdecompositionsdescribedin§2.Forthesedecompositions,thefollowingtwopropositions,forH(div)andH(curl),respectively,willbethekeyingredientsoftheanalysis.Theproofsofthesepropositions,unliketheproofofTheorems4.1and4.2,donotrequirethath≤cH.Also,sinceTheorems4.1and4.2areuna ectedbyscalingofthebilinearform,intheremainderofthepaperweassume,withoutlossofgenerality,thatρ=1.

dProposition4.3.Supposethatu∈Vhandthatu PHu∈Vhhasthediscrete

Helmholtzdecomposition

du PHu=gradhsh+curlqh,

forsomesh∈Shandqh∈curlhVh.Then

dκ gradhsh ≤cH u PHu Λd,d qh ≤cH u PHu .

cProposition4.4.Supposethatp∈Qhandthatp PHp∈Qhhasthediscrete

Helmholtzdecomposition

cp PHp=gradwh+curlhvh,

forsomewh∈Wh/Randvh∈Vhwithdivvh=0.Then

cp , wh ≤cH p PHcp Λc.κ curlhvh ≤cH p PH

Theproofoftheseresultsrequiresaseriesofintermediateresultsandwillbegiveninthenextsection.WenowshowhowthesepropositionsmaybeusedtoestablishTheorems4.1and4.2.

dProofofTheorem4.1.Sincev∈(I PH)Vh,itfollowsfromProposition4.3andthe

boundedre nementhypothesisthatvadmitsadiscreteHelmholtzdecomposition

+curlq,v=v