MULTIGRID IN H(div) AND H(curl)(7)

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)7

ThesediscreteHelmholtzdecompositionsareorthogonalinL2andinH(div)forthe rstandH(curl)forthesecond.

Remark.Ifweweretoconsiderthecasewhereessentialboundaryconditionsare h=Wh∩H 1,Q h=Qh∩H (curl),imposed,theappropriatespaceswouldbeW h=Vh∩H (div),andthecorrespondingexactsequencewouldbeandV

gradcurl div h h 0 →W →Q →Vh →Sh/R →0.

3.Abstractmultigridconvergences.Inthissection,werecallsomestandardresultsformultigriditerations,whichwillbethestartingpointforouranalysisofmultigridmethodsforthelinearsystemsofequationsdiscussedintheprevioussection.Suppose

X1 X2 ··· XJ=X

isasequenceof nitedimensionalsubspacesofaHilbertspace,Y,andΛ:X×X→Risasymmetricpositivede nitebilinearform.Foreachj,wede neΛj:Xj→Xjby(Λjx,y)=(Λx,y)forallx,y∈Xj.Ourgoalistheconstructionofane cientmultigriditerationtosolveorpreconditionequationsoftheformΛJx=f.LetMj:X→XjdenotetheY-orthogonalprojection,Pj:X→XjtheΛ-orthogonalprojection,andRj:Xj→Xjalinearoperator(thesmoother).Foreachj,wede neanY-symmetricoperatorΘj:Xj→XjbythestandardmultigridV-cycle1recursionwithm≥1smoothings.Thatis,wesetΘ1=Λ andforj>1and1f∈Xj,wede neΘjf=y2m+1where

y0=0∈Xj,

yi=yi 1+Rj(f Λjyi 1),

ym+1=ym+Θj 1Mj 1(f Λjym),

yi=yi 1+Rj(f Λjyi 1),i=m+2,m+3,...,2m+1.

ThenΘJistheV-cyclepreconditionerforΛJ.Thefollowingtheoremgivescondi-tionsonthesmoothersRjwhichensureconvergenceofthemultigridV-cycle(cf.,

[3],[4,Theorem3.6],or[2,Theorem5.1]).

Theorem3.1.Supposethatforeachj=1,2,...,J,thesmootherRjisY-symmetricandpositivesemide niteandsatis estheconditions

Λ([I RjΛj]x,x)≥0,

and 1(Rjx,x)≤αΛ(x,x),i=1,2,...,m,x∈Xj,x∈(I Pj 1)Xj,

whereαissomeconstant.Then

0≤Λ([I ΘJΛJ]x,x)≤δΛ(x,x),

whereδ=α/(α+2m).

Hence,themultigriderroroperatorI ΘJΛJisapositivede nitecontractionwithnormatmostδ<1independentofJanddecreasinginm,andtheprecondi-tionedoperatorΘJΛJhaseigenvaluesbetween1 δand1.x∈X,