MULTIGRID IN H(div) AND H(curl)(8)

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

8DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

ToobtainsmootherswhichsatisfytheconditionsofTheorem3.1,weconsideradditiveandmultiplicativeSchwarzoperators.Todescribethese,weassumethatkforeachj,therearespacesXj Xjsuchthateachx∈Xjcanbewrittenin Kktheformk=1xk,withxk∈Xj.LettingPjkdenotetheΛ-projectionoperator

kontothespaceXj,wecanthende netheunscaledadditiveSchwarzsmootherby ak 1aRj=KandthenthesmootherRj=ηRj,whereηisascalingfactor.k=1PjΛmWealsodenotebyRjtheusualmultiplicativeSchwarzsmootherassociatedwith

kmthespacesXj,i.e.,forf∈Xj,Rjf:=x2K,where

x0=0,

1xk=xk 1 Pjk(xk 1 Λ

jf),k=1,...,K,

k=K+1,...,2K.1xk=xk 1 Pj2K+1 k(xk 1 Λ

jf),

ThefollowingtheoremgivesconditionsonthedecompositionsoftheXjunderwhichtheSchwarzsmoothersleadtoaconvergentmultigriditeration.

Theorem3.2.Supposethat

K 1/2 K 1/2K K Λ(xk,yl) ≤β,Λ(xk,xk)Λ(yl,yl)

k=1l=1(3.1)k=1l=1

klxk∈Xj,yl∈Xj,

and

(3.2)infK kxk∈Xjk=1 x=xkΛ(xk,xk)≤γΛ(x,x),x∈(I Pj 1)Xj,

forsomeconstantsβ>0,γ>0.Then,

a(i)Ifη≤1/β,thescaledadditivesmoothersRj=ηRjsatisfythehypotheses

ofTheorem3.1withα=γ/η.

m(ii)ThemultiplicativesmoothersRj=RjsatisfythehypothesesofTheorem3.1

withα=β2γ.

Resultsofthistypecanbefoundinmanyplaces,forexamplein[4,Chapters3and5],[5],[13,Chapter5],and[15].Thereforewemerelysketchaproofhere.The rsthypothesisofTheorem3.1fortheadditivesmootherfollowsfrom(3.1)withxk=yk=PjkxandSchwarz’sinequality.Itiswellknownthattheleft

a 1handsideof(3.2)ispreciselyequalto(Rjx,x)(cf.equation(2.1)of[2]).The

secondhypothesisofTheorem3.1followsdirectlyfortheadditivesmoother.Forthemultiplicativesmoother,the rsthypothesisfollowsfromtheidentityΛ([I RjΛj]x,x)=Λ(Ex,Ex)whereE=(I PjK)(I PjK 1)···(I Pj1).Thesecond

amhypothesisisaconsequenceoftheinequality(Rjx,x)≤β2(Rjx,x),whichisjust

Corollary4.3of[2],usingtheargumentgivenattheendof§5ofthatpaper.