MULTIGRID IN H(div) AND H(curl)(15)

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)15

Hence, q ≤cH curlqh .Combiningthiswith(5.10),weobtain

d qh ≤cH curlqh ≤cH u PHu .

Thiscompletestheproofofthesecondestimateoftheproposition.

Sincethe rstestimateisvacuousifκ=0,weassumeκ>0.SinceΛdhmaps

1gradhShontoitself,wehavevh=(Λdgradhsh∈gradhSh.De ningvH∈h)

VHasinLemma5.1,wehave

2222 vh vH 2≤cH( divv +κ graddivv )dhhhΛh

≤cH2κ 2( vh 2+2κ2 divvh 2+κ4 gradhdivvh 2)

22 2=cH2κ 2 Λd gradhsh 2.hvh =cHκ

Hence,

dd gradhsh 2=Λd(gradhsh,vh)=Λd(u PHu,vh)=Λd(u PHu,vh vH)

dd≤ u PHu Λd vh vH Λd≤cHκ 1 u PHu Λd gradhsh .

WenowproveLemma5.2,fromwhichProposition4.4willfolloweasily.TheproofissubstantiallymoreinvolvedthanthatofLemma5.1,becausetheerrorestimate q qH ≤ q ΠQHq isnotvalid(re ectingthelackofthecommutativity

ZproperycurlΠQ

h=Πhcurl).

ProofofLemma5.2.Thelemmadoesnotinvolvetheparameterκ.Soasnottointroduceadditionalnotation,thenotationΛdisusedinthisprooftodenotedtheunweightedinnerproductinH(div)(κ=1),andPHisusedtodenotethecorrespondingorthogonalprojection.Z2SincecurlqH=ΠZHcurlqhwhereΠHistheLprojectionontoZH,weobvi-ouslyhave

(5.11) curlqH ≤c curlqh .

De ne(q,z)bytheboundaryvalueproblem(5.4)withfreplacedbycurlqh.SinceqhisthemixedapproximationofqinQhandcurlq∈Vh,weareabletouse(5.6)toestimateq qh.WhileqHisthemixedapproximationofqinQH,itisnottruethatcurlq∈VH,sowecannotestimateq qHinthesameway.Thereforewe

¯,z¯)by(5.4)withfreplacedbycurlqH.(Theanalogouscomplicationdidde ne(q

¯andψ=z z¯,weobtainnotariseintheproofofLemma5.1.)Settingφ=q q

curlψ=φ, ψ 1≤c φ ,

¯)=curl(qh qH),curlφ=curl(q q

φ 1≤c curlqh +c curlqH ≤c curlqh ,

¯ qH ≤cH curlqh , φ (qh qH) ≤ q qh + q

whereinthelastestimatewehaveused(5.6),(2.4),and(5.5)twice,andthen(5.11).