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Adaptive finite element methods: abstract framework and applications

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Niveau: Supérieur, Doctorat, Bac+8
Adaptive finite element methods: abstract framework and applications Serge Nicaise?, Sarah Cochez-Dhondt† June 25, 2008 Abstract We consider a general abstract framework of a continuous elliptic problem set on a Hilbert space V that is approximated by a family of (discrete) problems set on a finite- dimensional space of finite dimension not necessarily included into V . We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection- reaction-diffusion problems approximated by conforming P1 finite elements or by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented. Key Words A posteriori estimator, Adaptive FEM, Discontinuous Galerkin FEM. AMS (MOS) subject classification 65N30; 65N15, 65N50, 1 Introduction The convergence of adaptive algorithms for elliptic boundary value problems approximated by a conforming FEM started with the papers of Babuska and Vogelius [6] in 1d and of Dorfler [13] in 2d. Since this time some improvements have been proved in order to take into account the data oscillations [21, 22, 20] or to prove optimal arithmetic works [8]. On the other hand, the discontinuous Galerkin method becomes recently very popular and is a very efficient tool for the numerical approximation of reaction-convection-diffusion problems for instance.

  • adaptive algorithm

  • diffusion problem

  • discontinuous galerkin

  • †universite de valenciennes et du hainaut cambresis

  • techniques de valenciennes

  • reaction-convection-diffusion problems

  • local error


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Language English
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Adaptive
finite
element methods: abstract and applications
Serge Nicaise,Sarah Cochez-Dhondt
June 25, 2008
Abstract
framework
We consider a general abstract framework of a continuous elliptic problem set on a Hilbert spaceVis approximated by a family of (discrete) problems set on a finite-that dimensional space of finite dimension not necessarily included intoV. We give a series of realistic conditions on an error estimator that allows to conclude that the marking strategy of bulk type leads to the geometric convergence of the adaptive algorithm. These conditions are then verified for different concrete problems like convection-reaction-diffusion problems approximated by conformingP1finite elements or by a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.
Key WordsA posteriori estimator, Adaptive FEM, Discontinuous Galerkin FEM. AMS (MOS) subject classification65N30; 65N15, 65N50,
1 Introduction
The convergence of adaptive algorithms for elliptic boundary value problems approximated by a conforming FEM started with the papers of Babuska and Vogelius [6] in 1d and of D¨orer[13]in2d.Sincethistimesomeimprovementshavebeenprovedinordertotake into account the data oscillations [21, 22, 20] or to prove optimal arithmetic works [8]. On the other hand, the discontinuous Galerkin method becomes recently very popular and is a very efficient tool for the numerical approximation of reaction-convection-diffusion problems for instance. Some a posteriori error analysis were performed recently, let us -icSsedtevsrUinetesHaduauinamtCe´tiaVedcnelnneiCNRS2956,Institurbe´is,sALAM,VRF ences et Techniques de Valenciennes, F-59313 - Valenciennes Cedex 9 France, email: Serge.Nicaise@univ-valenciennes.fr NRRC95S2AM,L,FAVedtuicSsnI,6titsnesetduHalencienbm´rsesiiaantuaCVede´tisrevinU-ences et Techniques de Valenciennes, F-59313 - Valenciennes Cedex 9 France, email: Sarah.Cochez@univ-valenciennes.fr
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