MULTIGRID IN H(div) AND H(curl)(13)

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)13

22whereΠZh:L→ZhistheLprojection,andqhaloneischaracterizedasthe

uniquefunctionincurlhVhsatisfyingthelatterequation.Atthispoint,anessentialdi erencebetweenthemixedapproximationof(5.4)and(5.1)arises.Itisnottrue

ZZthatcurlΠQ

hq=Πhcurlqforallsmoothfunctionsq(sinceΠhdoesnotcoincide

withΠVh,evenwhenappliedtoirrotational elds).Asaresult,itisnotingeneraltruethat q qh ≤ q ΠQ

hq .However,thisestimateistrueinthespecialcase

thatf∈Zh,i.e.,

(5.6) q qh ≤ q ΠQ

hq ,q∈H1suchthatcurlq∈Vh.

Indeed,inthiscase

VZcurlΠQ

hq=Πhcurlq=curlq=Πhcurlq=curlqh,

soΠQ

hq qhiscurl-free.Itthenfollowsdirectlyfromthede ningequationsofthemixedmethodthat(q qh,ΠQ

hq qh)=0,whichgives(5.6).

Noticethatthehypothesiscurlq∈Vhisalsowhatisneededfortheapproxima-tionestimate(2.4).Combining(5.6),(2.4),andthecontinuousinf–supcondition,wegetthediscreteinf–supcondition,

infsup(curlq,z)

z∈Zhq∈Qh