MULTIGRID IN H(div) AND H(curl)(12)
时间:2025-07-09
时间:2025-07-09
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
12DOUGLASN.ARNOLD,RICHARDS.FALK,ANDRAGNARWINTHER
overL2×H(div).ThisisamixedvariationalformulationoftheDirichletboundaryvalueproblem
(5.1)v=grads,divv=fin ,s=0on .
Themixed niteelementapproximation(sh,vh)to(s,v)istheuniquecriticalpointofLdoverSh×Vh.Itisdeterminedbytheequationsvh=gradhsh,divvh=ΠShf,andvhaloneischaracterizedastheuniquefunctioningradhShsatisfyingthelatterequation.Abasicestimateformixedmethodsis
(5.2) v vh ≤ v ΠVhv ,v∈H1,
SwhichisaconsequenceofthecommutativitypropertydivΠVh=Πhdiv.Fromthe
propertiesoftheoperatorΠVhonealsoeasilyderivestheinf–supcondition:
s∈Shv∈Vhinfsup(divv,s)
2 q 2 (curlq,z)+(f,z).
Thiscorrespondstotheboundaryvalueproblem
(5.4)q=curlz,curlq=f,divz=0in ,z×n=0on .
Forthisproblemwehaveq,z∈H1and
(5.5) q 1≤c f , z 1≤c q .
Indeed,sincethenormalcomponentofq=curlzisthetangentialdivergenceofz×n,whichvanisheson ,itfollowsthatq·n=0on .TheestimatesonqandzarethengiveninTheorems2.1and2.2of[7],respectively.
Themixed niteelementapproximation(zh,qh)istheuniquecriticalpointofLcoverZh×Qh.Itisdeterminedbytheequationsqh=curlhzh,curlqh=ΠZhf,
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