MULTIGRID IN H(div) AND H(curl)(5)

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)

5

Fig.1.DegreesoffreedomforthespacesWh,Qh,Vh,andShinthe

lowestordercasek=0.

andanalogouslywithQhreplacedbyWh,Vh,orSh.Then

Wh=

Qh=

(2.1)Vh=

Sh= vWh,v∈Vhv∈Vh QvhvVhvSh===v∈Vh

v∈Vh e∈Eh Qeh,eVheShe∈Eh e∈Eh ==f∈Fh f∈FhfSh fVh,=T∈Th TSh.

(2.2)ForeachofthesedecompositionsthereisacorrespondingestimateonthesumofthesquaresoftheL2normsofthesummands.Forexample,wecandecomposean arbitraryelementq∈Qhasq=e∈Ehqewithqe∈Qehsothattheestimate

e∈Eh qe 2≤c q 2

holdswithcdependingonlyontheshaperegularityofthemesh.Weremarkalso thatethedecompositionsnotstatedinfactdon’thold.Forexample,Wh=e∈EhWh.

Theproofofthesedecompositionsandthecorrespondingestimatesallfollowthesamelines.Forexample,toprovetheedge-baseddecompositionofQhandtheestimate(2.2),wenotethatthedegreesoffreedomofthespaceQhdeterminea canonicaldecompositionofanarbitraryelementq∈Qhasq=qξwherethesumrunsoverallthedegreesoffreedomofQh,andqξistheelementofQhwithalldegreesoffreedomotherthanξsetequaltozero.Astandardscalingargumentthenimpliesthat qξ ≤c q L2(suppqξ).Nowtoeachdegreeoffreedomξ,wemayassignanedgeeofthemeshsuchthatsuppqξ eh(foranedge-baseddegreeoffreedom,chooseetobethatedgeandforaface-ortetrahedron-baseddegreeoffreedom,chooseetobeanyedgecontainedinthefaceortetrahedron).Combiningthecorrespondingqξgivesthedesireddecomposition.

Thesedegreesoffreedomspeci edforthespacesWh,Qh,Vh,andShdeter-QVSmineinterpolationoperatorsΠW,Πhh,Πh,andΠhmappingfunctionswhichare