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A new approach to Richardson extrapolation in the finite element method for second order elliptic problems


Authors: M. Asadzadeh, A. H. Schatz and W. Wendland
Journal: Math. Comp. 78 (2009), 1951-1973
MSC (2000): Primary 65N15, 65N30, 35J25
DOI: https://doi.org/10.1090/S0025-5718-09-02241-8
Published electronically: February 11, 2009
MathSciNet review: 2521274
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Abstract: This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in $ \mathbb{R}^N$, $ N \ge 2$. The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given.


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Additional Information

M. Asadzadeh
Affiliation: Department of Mathematics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden
Address at time of publication: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
Email: mohammad@chalmers.se, asadzadeh@math.cornell.edu

A. H. Schatz
Affiliation: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, New York 14853
Email: schatz@math.cornell.edu

W. Wendland
Affiliation: Institute for Applied Analysis and Numerical Simulations, University of Stuttgart, Pfaffenwaldring 57, D-750550, Germany
Email: wendland@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S0025-5718-09-02241-8
Keywords: Richardson extrapolation, local estimates, asymptotic error expansion inequalities, similarity of subspaces, scalings, finite element method, elliptic equations
Received by editor(s): November 21, 2007
Received by editor(s) in revised form: October 11, 2008
Published electronically: February 11, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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