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

Asadzadeh, M.; Schatz, A.; Wendland, W.

doi:10.1090/S0025-5718-09-02241-8

A new approach to Richardson extrapolation in the finite element method for second order elliptic problems
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by M. Asadzadeh, A. H. Schatz and W. Wendland;

Math. Comp. 78 (2009), 1951-1973

DOI: https://doi.org/10.1090/S0025-5718-09-02241-8

Published electronically: February 11, 2009

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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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