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A multiplicative Schwarz adaptive wavelet method for elliptic boundary value problems


Authors: Rob Stevenson and Manuel Werner
Journal: Math. Comp. 78 (2009), 619-644
MSC (2000): Primary 65N55, 65T60, 41A25
DOI: https://doi.org/10.1090/S0025-5718-08-02186-8
Published electronically: November 13, 2008
MathSciNet review: 2476554
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Abstract: A multiplicative Schwarz overlapping domain decomposition method is considered for solving elliptic boundary value problems. By equipping the relevant Sobolev spaces on the subdomains with wavelet bases, adaptive wavelet methods are used for approximately solving the subdomain problems. The union of the wavelet bases forms a frame for the Sobolev space on the domain as a whole. The resulting method is proven to be optimal in the sense that, in linear complexity, the iterands converge with the same rate as the sequence over $ N \in \mathbb{N}$ of the best approximation from the span of the best $ N$ frame elements. Numerical results are given for the method applied to Poisson's equation.


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Additional Information

Rob Stevenson
Affiliation: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Email: rstevens@science.uva.nl

Manuel Werner
Affiliation: Fachbereich 12 Mathematik und Informatik, Philipps–Universität Marburg, Hans–Meerwein–Strasse, Lahnberge, D–35032, Marburg, Germany
Email: werner@mathematik.uni-marburg.de

DOI: https://doi.org/10.1090/S0025-5718-08-02186-8
Keywords: Elliptic boundary value problems, wavelets, frames, adaptivity, best $N$-term approximation, multiplicative Schwarz method, domain decomposition
Received by editor(s): March 6, 2008
Published electronically: November 13, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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