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Adaptive finite element methods for elliptic problems: Abstract framework and applications

Published online by Cambridge University Press:  04 February 2010

Serge Nicaise  and
Sarah Cochez-Dhondt
Serge Nicaise
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr; Sarah.Cochez@univ-valenciennes.fr
Sarah Cochez-Dhondt
Affiliation:
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr; Sarah.Cochez@univ-valenciennes.fr
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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 notnecessarily included into V. We give a series of realisticconditions on an error estimator that allows to conclude that themarking strategy of bulk type leads to the geometric convergenceof the adaptive algorithm. These conditions are then verified fordifferent concrete problems like convection-reaction-diffusionproblems approximated by a discontinuous Galerkin methodwith an estimator of residual type or obtained by equilibratedfluxes. Numerical tests that confirm the geometric convergence arepresented.

Keywords

A posteriori estimatoradaptive FEMdiscontinuous Galerkin FEM
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 485 - 508
Copyright
© EDP Sciences, SMAI, 2010

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