MULTIGRID IN H(div) AND H(curl)(17)

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)17

asdesired.

Itremainstoprove(5.14).Foru∈curlQh,weusethediscreteHelmholtzdecompositiontowrite

d(I PH)u=curlp+gradhs,s∈Sh,p∈curlhVh,

andthentowrite

d(I PH)curlp=curlm+gradhr,r∈Sh,m∈curlhVh.

Fromthe rstestimateofProposition4.3andthefactthatuisdivergence-free,wehavethat

d gradhs ≤cH u PHu H(div)≤cH u H(div)=cH u .

Againusingthevanishingofdivu,weobtain

dd]curlp,u) curlp 2=Λd(curlp,[I PH]u)=Λd([I PH

d=([I PH]curlp,u)=(curlm,u)=(m,curlhu).

FromthesecondestimateofProposition4.3wethenget

d m ≤cH (I PH)curlp H(div)≤cH curlp H(div)=cH curlp .

Hence, curlp ≤cH curlhu .Finally,

du ≤ curlp + gradhs ≤cH( u + curlhu ), u PH

which,togetherwith(5.7),establishes(5.14).

References

1.C.Amrouche,C.Bernardi,M.Dauge,andV.Girault,Vectorpotentialsinthree-dimensionalnonsmoothdomains,Math.MethodsAppl.Sci.(1998)(toappear).

2.D.N.Arnold,R.S.Falk,andR.Winther,PreconditioninginH(div)andapplications,p.66(1997),957–984.

3.D.BraessandW.Hackbusch,AnewconvergenceproofforthemultigridmethodincludingtheV-cycle,SIAMJ.Numer.Anal.20(1983),967–975.

4.J.H.Bramble,Multigridmethods,PitmanResearchNotesinMathematicsSeries294(1993).

5.M.DryjaandO.B.Widlund,SchwarzmethodsofNeumann–Neumanntypeforthree-dimen-sionalelliptic niteelementproblems,Comm.PureAppl.Math48(1995),121–155.

6.Z.Cai,C.I.Goldstein,andJ.E.Pasciak,Multileveliterationformixed niteelementsystemswithpenalty,put.14(1993),1072–1088.

7.V.Girault,Incompressible niteelementmethodsforNavier–Stokesequationswithnonstan-dardboundaryconditionsinR3,p.51(1988),55–74.

8.V.GiraultandP-A.Raviart,FiniteElementMethodsforNavier–StokesEquations,Springer-Verlag,Berlin,1986.

9.W.Hackbush,MultigridMethodsandApplications,Springer-Verlag,Berlin,1986.

10.R.Hiptmair,MultigridmethodforH(div)inthreedimensions,ETNA6(1997),133–152.11.