LEAST-SQUARES MIXED FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

作者: 刊名:Journal of Computational Mathematics 上传者:陈岩

【摘要】Two least-squares mixed finite element schemes are formulated to solve the initial-boundary value problem of a nonlinear parabolic partial differential equation and the convergence of these schemes are analyzed.

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1.IntroductionAlargenumberofphysicalphenomenaaremodeledbypartialdifferelltialequationsorsystemsofparabolictypeinanevolutionaryoreIliptictypeatsteadystate.Itisfrequentlythecasethatagoodapproximationofsomefunctionofthegradientofthesolutiontothedifferentialequation(whichmayrepresent,forexample,avelocityfieldorelectricfield)isatleastasimportantasanapproximationofthesolutionitself(whichmayrepresent,respectivelyapressureoranelectricpotential).Manymixedelementmethodscomputesimultaneouslythesolutionandthegradielltofthesolutionwiththesameorhigherorderofaccuracythanthesolutionitself.Themixedmethodsweredescribedandanalyzedbymanyauthors.IthasbeenobservedthatinmanycasesmixedfiniteelementmethodsgivebetterapproximationsfOrthefltixvariablethanclassicalGalerkinmethods.However,amixedformulationismoredifficulttobehandledand,ingeneral,ismoreexpensivefromacomputationalpoifltofviewbecauseitlosespositivedefiniteproperty.Recently,therehasbeenanincreasinginterestintheapplicationsofleast-squaresfiniteelementalgorithmstovariousproblemssteadyorevo1utionaryManyworksonleast-squaresfiniteelementschemesandtheiraPplicationstovariousboundaryvalueproblemsofellipticequationsorsystemshavebeendoneandsomesystematictheoriesonellipticityanderrorestimateshavebeenalsoestabished,e.g.,see[2],[3],[7]-[12],[15]-[18l,[221and[23].Inrecentyears,theleast-squaresfiniteelementmethodshavebeenextendedtotime-dependeatproblems,e.g.,see[13],[21]and[25]tandseveralnumericalresultsshowedthatleast-squaresfiniteelementmethodsarealsoverye

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