MULTIGRID IN H(div) AND H(curl)(16)

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

16DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

Weestimate φ usingthesamedualityargumentweusedtoestimate q intheproofofProposition4.3.SinceΠVHψ∈ZH(whichfollowsfromthecommutativitySrelationdivΠVH=ΠHdiv),andcurl(qh qH)⊥ZH,we nd

φ 2=(φ,curlψ)=(curlφ,ψ)=(curl[qh qH],ψ)

=(curl[qh qH],ψ ΠVHψ)≤cH curl(qh qH) ψ 1

≤cH curl(qh qH) φ .

Thisimpliesthat φ ≤cH curl(qh qH) ≤CH curlqh ,andsoweobtainthe rstestimateofthelemma.

Itremainstoprovethesecondestimate.Forthisestimate,too,wecannotsimplyusetheanalogueoftheargumentthatestablishedthesecondestimateofProposition4.3.Thistimetheproblemcanbetracedtothefailureofthecom-QmutativitypropertyΠZcurl=curlΠHH,eventhoughtheanalogousproperty

VΠSHdiv=divΠHisvalid.Insteadweshallderivetheestimatebyestablishingthe

followingthreefacts:

d(5.12)curlqh curlqH=(I PH)curlqh+gradHsH,forsomesH∈SH,

(5.13)

(5.14)d)curlqh , gradHsH ≤c (I PHd u PHu ≤cH curlhu ,u∈curlQh.

Thedesiredestimatefollowsbytakingu=curlqhin(5.14)andusing(5.12)and(5.13).

The rststatementfollowsfromtheequations

(curlqH,curlrH)=(curlqh,curlrH)=Λd(curlqh,curlrH)

dd=Λd(PHcurlqh,curlrH)=(PHcurlqh,curlrH),rH∈QH.

dcurlqhandToprove(5.13),wenotefromtheHelmholtzdecompositionofPHd,thatforanyvH∈VH,thede nitionofPH

d(divgradHsH,divvH)=(divPHcurlqh,divvH)

(5.15)dd=Λd(PHcurlqh,vH) (PHcurlqh,vH)

dd=(curlqh,vH) (PHcurlqh,vH)=([I PH]curlqh,vH).

Now

gradHsH 2= (divgradHsH,sH)≤ divgradHsH sH

≤c divgradHsH gradHsH ,

bythediscretePoincar´einequality(5.3).Thus gradHsH ≤c divgradHsH ,andtakingvH=gradHsHin(5.15),weget

d gradHsH 2≤c divgradHsH 2=c([I PH]curlqh,gradHsH)

d≤c (I PH)curlqh gradHsH ,