MULTIGRID IN H(div) AND H(curl)(11)

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)11

∈gradhShandq∈Qhsatisfytheboundswherev

(4.5) ≤ v , v ≤ch v Λd,κ v q ≤ch v .

e∈Eh =Followingthediscussionof§2,wecanwritev

(4.6)

Thenv=

(4.7) ≤c v , ve22vandq= ee∈Ehe∈Eh qe 2≤c q 2. e∈Ehqewithe∈Eh e+curlqe.Moreover,usinganinverseinequality,vewhereve:=v= e2 e 2( v+ curlq )dΛ

e∈Eh ve 2Λde∈Eh

≤c

e∈Eh e 2+h 2 qe 2],[(1+κ2h 2) v

andthetheoremfollowsfrom(4.5)–(4.7).

cProofofTheorem4.2.Sinceq∈(I PH)Qh,itfollowsfromProposition4.4and

theboundedre nementhypothesisthatqisgivenby

+gradw,q=q

∈curlhVhandw∈Whsatisfytheestimateswhereq

≤ q , q

v ≤ch q Λc,κ q w ≤ch q .

=v∈Vhq andw=v∈Vhwv,andsettingqv=q v+gradwv,weWritingqcompletetheproofasfortheprecedingtheorem.

Remark.TheproofofTheorem4.1appliesalmostwithoutmodi cationifthede- ecompositionVh=e∈EhVhisreplacedbyeithertheseconddecompositionin(4.1)orthedecompositionin(4.2).Similarly,theproofofTheorem4.2appliestothedecompositionin(4.4)aswell.Itisalsoclearwhywecannotusetheface-baseddecompositionofVhinTheorem4.1,sincetheproofwouldrequireacorrespondingface-baseddecompositionofQh,whichdoesnotexist.

5.Two-levelestimatesformixed niteelements.InthissectionweprovePropositions4.3and4.4.Ourproofsarebasedonestimatesfortheapproximationofdiscretelyirrotationalvector eldsinVhanddiscretelysolenoidalvector eldsinQhbydiscretelyirrotationalandsolenoidal eldsinVHandQH,respectively.Thesetwo-levelapproximationresults,inturn,relyonestimatesformixed niteelementmethodsbasedonH(div)andH(curl).Webeginthissectionwithadiscussionofsuchmethods.

FortheH(div)case,letf∈L2andde ne(s,v)astheuniquecriticalpoint(asaddle)of1Ld(s,v):=