MULTIGRID IN H(div) AND H(curl)(14)

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

14DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHERProofofLemma5.1.De ne(v,s)from(5.1)withf=divvh.ThenvhandvHarethemixedapproximationstovinVhandVH,respectively.Applying(5.2),(2.3),and2-regularityfortheDirichletproblem,weobtain

v vH ≤ v ΠVHv ≤cH v 1≤cH divvh ,

and,similarly, v vh ≤ch divvh .The rstestimatethusfollowsfromthetriangleinequality.

Nextweprovethatforanyrh∈Sh,

(5.8) rh ΠSHrh ≤cH gradhrh .

Inparticular,wemaytakerh=divvhinthisestimate,toget

divvh divvH ≤cH gradhdivvh .

Toprove(5.8),wede neafunctionuwhichsatis es

divu=rh ΠSHrh,

Then

2SSS rh ΠSHrh =(divu,rh ΠHrh)=(divu,Πhrh ΠHrh)

VVS=([ΠSh ΠH]divu,rh)=(div[Πh ΠH]u,rh)

VVV=([ΠVh ΠH]u,gradhrh)≤( Πhu u + u ΠHu ) gradhrh u 1≤ rh ΠSHrh .

≤cH u 1 gradhrh ≤cH rh ΠSHrh gradhrh ,

whichimplies(5.8).

ProofofProposition4.3.Thepropositiondirectlygeneralizesthecorrespondingtwo-dimensionalresult,Lemma3.1of[2].Theproofoftheboundongradhshisentirelyanalogoustotheargumentin[2],buttheboundforqhrequirestheuseofamorecomplicateddualityargument.First,observethat

(5.9)d(curlqh,curlr)=Λd(u PHu,curlr)=0,r∈QH.

De ne(q,z)asin(5.4)withfreplacedbycurlqh.Thenqh∈Qhisthemixedapproximationtoq,andhence,by(5.6),(2.4),and(5.5),

(5.10) q qh ≤ q ΠQ

hq ≤ch q 1≤ch curlqh .

VSincedivz=0,divΠVHz=0,andsoΠHz∈curlQH.Wemaythereforeapply

(5.9),(2.3),and(5.5)toobtain

q 2=(q,curlz)=(curlq,z)=(curlqh,z)=(curlqh,z ΠVHz)

≤cH z 1 curlqh ≤cH q curlqh .