MULTIGRID IN H(div) AND H(curl)(6)

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

6DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER

su cientlysmooththattherequiredfunctionvaluesandmomentsexistintothesubspaces.Speci cally,ΠWhisastandardinterpolationoperatorandisde nedon

22continuousfunctions,andΠShistheL-projectionoperator,de nedonallLfunc-

1tions.AsthedomainforΠVh,wecanchooseH.Moreover,astandardargument

basedontheBramble–Hilbertlemmaandscalinggivestheerrorestimate(2.3) v ΠVhv ≤ch v 1,v∈H1.

Becauseofthedependenceonedgemoments,thesituationismorecomplicated

1fortheoperatorΠQ

h.ItisboundedonthespaceofHvector eldswhosecurl

belongstoLp,forany xedp∈(2,∞].ThisfollowsfromLemma4.7of[1]andtheSobolevembeddingtheorem.Inparticular,itisde nedforH1vector eldswhosecurlbelongstoVh.Moreoverwehave

(2.4) q ΠQ

hq ≤ch q 1,q∈H1suchthatcurlq∈Vh.

Toshowthis,wefollow[12].Firstconsiderthecasewherethemeshconsistsof .LetQ andV denotethecorrespondingspacesandΠ QonlytheunitsimplexT ,ingtheequivalenceofnormsinV

Qq ≤c( q 1+ curlq L∞)≤c q 1 Π

.ABramble–Hilbertargumentthengives )suchthatcurlq ∈H1(T ∈Vforallq Qq )suchthatcurlq wherenowonlytheH1 Π ≤c|q |1forq ∈H1(T ∈V q

seminormappearsontherighthandside.Ifwescalethisestimatetoageneral withFa ne,usingtheappropriatecontravarianttransformsimplexT=F 1T

→(DF) (q F),andaddupoverallthesimplicesinthemesh,weget(2.4).q

Theinterpolationoperatorsalsosatisfythecommutativityproperties

VcurlΠQ

h=Πhcurl,SdivΠVh=Πhdiv,QgradΠWh=Πhgrad

whenappliedtosu cientlysmoothvector elds.Thesewell-knownrelationsfollowfromthede nitionsoftheinterpolationoperatorsandthetheoremsofGreenandStokes.dInadditiontotheseinterpolationoperators,wealsode nePhtobetheorthog-conalprojectionontoVhwithrespecttotheinnerproductinH(div)andPhtobetheorthogonalprojectionontoQhwithrespecttotheinnerproductinH(curl).AkeypropertyrelatingthespacesWh,Qh,Vh,andSh,isthatthefollowingsequenceisexact:

0 →Wh/R →Qh →Vh →Sh →0,

i.e.,thattherangeofeachoftheoperatorsinthesequencecoincideswiththenullspaceofthefollowingoperator.Itfollowsthatifwede negradh:Sh→VhastheL2adjointofthemap div:Vh→Sh,andcurlh:Vh→QhastheL2adjointofcurl:Qh→Vh,thenwehavethetwoorthogonaldecompositions:

Vh=curlQh⊕gradhSh,Qh=curlhVh⊕gradWh.gradcurldiv