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A Review of Unified A Posteriori Finite Element Error Control

Published online by Cambridge University Press:  28 May 2015

C. Carstensen ,
M. Eigel ,
R. H. W. Hoppe  and
C. Löbhard
C. Carstensen*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, Korea
M. Eigel*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany
R. H. W. Hoppe*
Affiliation:
Department of Mathematics, University of Houston, Houston TX 77204-3008, USA Institute of Mathematics, University of Augsburg, D-86159 Augsburg, Germany
C. Löbhard*
Affiliation:
Department of Mathematics, Humboldt Universität zu Berlin, D-10099 Berlin, Germany
*
Corresponding author.Email address:cc@math.hu-berlin.de
Corresponding author.Email address:eigel@math.hu-berlin.de
Corresponding author.Email address:hoppe@math.uni-augsburg.de
Corresponding author.Email address:loebhard@math.hu-berlin.de

Abstract

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This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations. In the abstract setting of mixed formulations, a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms. Notably, this can be done with an approach which is applicable in the same way to conforming, nonconforming and mixed discretisations. Subsequently, the unified approach is applied to various model problems. In particular, we consider the Laplace, Stokes, Navier-Lamé, and the semi-discrete eddy current equations.

Keywords

65N3065N1535J25A posteriorierror analysisfinite element methodnonconforming finite element methodmixed finite element methodadaptive algorithmPoisson equationLamé equationsStokes equationsMaxwell equationsunified a posteriori error analysisdiscontinuous Galerkinresidual estimator
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 4 , November 2012 , pp. 509 - 558
Copyright
Copyright © Global Science Press Limited 2012

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