MULTIGRID IN H(div) AND H(curl)(4)

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

4DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

applytheseresulttoobtaintheconvergenceofthestandardmultigridV-cycleforctheoperatorsΛdhandΛhusingappropriatelyde nedadditiveandmultiplicativeSchwarzsmoothers.Theproofhingesoncertaintwo-levelerrorestimatesformixedmethodsbasedontheRaviart–Thomas–NedelecandNedelecedgespaces.Theseestimatesarestatedandprovedin§5.

2.Finiteelementdiscretization.Wesupposethat isaboundedandconvexpolyhedroninR3andThameshof consistingofclosedtetrahedra.Weassumethatthemeshisshaperegularandquasi-uniform.Moreprecisely,theconstantsthatappearintheestimatesbelowmaydependontheshaperegularityconstant(themaximumratioofthediameterofanelementtothediameterofthelargestballcontainedintheelement)andthequasi-uniformityconstant(themaximumratioofthelargesttothesmallestelementdiameter)ofthemesh,butareotherwisemesh-independent.WedenotebyVh,Eh,andFhthesetsofvertices,edges,andfacesofthemesh,respectively.Forν∈Vh∪Eh∪Fh∪Thwede ne νννTh={T∈Th|ν T}, h=interior(Th).

Thus νhisthesubdomainof formedbythepatchofelementsmeetingν,andνistherestrictionofthemeshThto νThh.

Fixanintegerk≥0.Wethenrecallthefollowingspaces:

Wh:

Qh:

Vh:

Sh:continuouspiecewisepolynomialsofdegreeatmostk+1,theNedelecedgediscretizationofH(curl)ofindexk,theRaviart–Thomas–NedelecdiscretizationofH(div)ofindexk,arbitrarypiecewisepolynomialsofdegreeatmostk.

Tode nethesespaces,wespecifythecorrespondingpolynomialspacesusedoneachelementandthecorrespondingsetsofdegreesoffreedom.RestrictedtoatetrahedronT,theelementsofWhandShare,ofcourse,arbitraryelementsofPk+1(T)andPk(T),respectively,wherePk(T)denotesthespaceofpolynomialsofdegreeatmostkrestrictedtoT.TherestrictionsoftheelementsofVharefunctionsoftheformp(x)+r(x)xwithp∈Pk(T)andr∈Pk(T).TheelementsofQharefunctionsoftheformp(x)+r(x)withp∈Pk(T)andr∈Pk+1(T)suchthatr·x≡0.Thedegreesoffreedomforu∈Vhareoftwosorts.First,themomentsofu·noforderatmostkoneachfacef(morepreciselythefunctionalsthatassociatetouitsinnerproductinL2(f)witheachelementofabasisforPk(f));andsecond,themomentsofuofdegreek 1oneachtetrahedron.Thedegreesoffreedomofq∈Qhare(1)themomentsofq·soforderatmostkoneachedge,(2)themomentsofq×noforderatmostk 1oneachface,and(3)themomentsofqoforderatmostk 2oneachtetrahedron.ForShweuseasdegreesoffreedomthetetrahedralmomentsoforderatmostk.ForWh,weuse

(1)thevaluesatthevertices,(2)theedgemomentsoforderatmostk 1,(3)thefacemomentsoforderatmostk 2,and(4)thetetrahedralmomentsoforderatmostk 3.

Wenowconsiderthedecompositionofthesespacesassumsofspacessupportedinsmallpatchesofelements.De ne

¯νQνh={r∈Qh:suppr h},ν∈Vh∪Eh∪Fh∪Th,