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

Serge Nicaise; Sarah Cochez-Dhondt

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 3, page 485-508
  • ISSN: 0764-583X

Abstract

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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 a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.

How to cite

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Nicaise, Serge, and Cochez-Dhondt, Sarah. "Adaptive finite element methods for elliptic problems: Abstract framework and applications." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 485-508. <http://eudml.org/doc/250819>.

@article{Nicaise2010,
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 a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented. },
author = {Nicaise, Serge, Cochez-Dhondt, Sarah},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {A posteriori estimator; adaptive FEM; discontinuous Galerkin FEM; elliptic boundary value problems; a posteriori estimator; finite element method; convergence; continuous elliptic problems; Hilbert spaces; numerical examples; convergence; adaptive algorithm; Dirichlet boundary value problem; linear convection-diffusion-reaction problems; discontinuous Galerkin method},
language = {eng},
month = {4},
number = {3},
pages = {485-508},
publisher = {EDP Sciences},
title = {Adaptive finite element methods for elliptic problems: Abstract framework and applications},
url = {http://eudml.org/doc/250819},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Nicaise, Serge
AU - Cochez-Dhondt, Sarah
TI - Adaptive finite element methods for elliptic problems: Abstract framework and applications
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/4//
PB - EDP Sciences
VL - 44
IS - 3
SP - 485
EP - 508
AB - 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 a discontinuous Galerkin method with an estimator of residual type or obtained by equilibrated fluxes. Numerical tests that confirm the geometric convergence are presented.
LA - eng
KW - A posteriori estimator; adaptive FEM; discontinuous Galerkin FEM; elliptic boundary value problems; a posteriori estimator; finite element method; convergence; continuous elliptic problems; Hilbert spaces; numerical examples; convergence; adaptive algorithm; Dirichlet boundary value problem; linear convection-diffusion-reaction problems; discontinuous Galerkin method
UR - http://eudml.org/doc/250819
ER -

References

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