# 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

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topNicaise, 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 -

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