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 | References | Similar Articles | Additional Information

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 , . 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.