帶間斷系數(shù)二階橢圓問題的非協(xié)調(diào)有限元的多水平和區(qū)域分解預(yù)處理方法.pdf

1、南京師范大學(xué)博士學(xué)位論文帶間斷系數(shù)二階橢圓問題的非協(xié)調(diào)有限元的多水平和區(qū)域分解預(yù)處理方法姓名：王鋒申請學(xué)位級別：博士專業(yè)：基礎(chǔ)數(shù)學(xué)指導(dǎo)教師：陳金如20090515AbstractAbstractInthisthesis，underthegeneralframeworkoftheadditiveSchwarzmethods，wediscussefficientpreconditionersforseveralnonconformingfi

2、niteelementmethodsforsecondorderellipticpartialdifferentialequationswithpiecewisebutdiscontinuouscoe伍cientsFirstlyamultilevelpreconditionerfortheCrouzeixRaviartfiniteelementapproximationisproposedSincethefiniteelementspa

3、cesdefinedonnestedgridsarenonnested，weightedintergridtransferoperators，whicharestableundertheweightedL2norm，areintroducedtoexchangemessagesbetweendifferentmeshesAlthoughtheconditionnumberofthepreconditionedsystemdependso

4、nthejumpinthecoefficient，byanalyzingtheeigenvaluedistributionofthepreconditionedsystem，wegetthatexceptafewsmalleigenvalues，alltheothereigenvaluesareboundedbelowandabovenearlyuniformlywithrespecttothejumpandmeshsizeAsares

5、uR，theeffec—tiveconditionnumberofthepreconditionedsystemisrobustandquasi—optimalwithrespecttothemeshsizeThen，theasymptoticconvergencerateofthepreconditionedconjugategradientmethodis1一南Secondlyusingthepiecewisebilinear／tr

6、ilinearQ1elementspaceasacoarsespace，weproposeandanalyzeatwoleveladditiveSchwarzpreconditionerforthePlnonconformingquadrilateralfiniteelementmethodSimilarlyitisprovedthattheeffectiveconditionnumberofthepreconditionedsyste

7、misindependentofthejumpandquasioptimalwithrespecttOthecoarsemeshsize，ie，theeffectiveconditionnumberis0(1log(H0，fH沁，whereHandHodenotethecoarsemeshsizeandthediametersofthesubdommnscarvedbythediscontinuouscoefficientsrespec

8、tivelyFinallyweconsiderthemortartyperotatedQ1finiteelementmethodForthismethod，ouranalysisisvalidforrectangularorL—shapeddomains，whicharepattifionedbyrectangularsubdomainsandmeshesDuetothedifficultycausedbythemortarcondit

9、ion，wedesignanauxiliaryproblemwhichisequivalenttotheorigi—haloneUsingspecialdiscreteharmonicspacesascoarsespaces，severaldomaindecompositionprecoditionersarepresentedfortheauxiliaryproblem，andareconsequentlyappliedtOtheor