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Optimized Schwarz Methods without Overlap for the Helmholtz Equation

Authors: Martin J. Gander, Frédéric Magoulès, and Frédéric NatafAuthors Info & Affiliations
https://doi.org/10.1137/S1064827501387012
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Abstract

The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.

MSC codes

  1. 65F10
  2. 65N22

Keywords

  1. optimized Schwarz methods
  2. domain decomposition
  3. preconditioner
  4. iterative parallel methods
  5. acoustics

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Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Volume 24 • Issue 1 • January 2002
Pages: 38 - 60
DOI: 10.1137/S1064827501387012
ISSN (online): 1095-7197

Copyright

Copyright © 2002 Society for Industrial and Applied Mathematics.

History

Published online: 25 July 2006

MSC codes

  1. 65F10
  2. 65N22

Keywords

  1. optimized Schwarz methods
  2. domain decomposition
  3. preconditioner
  4. iterative parallel methods
  5. acoustics

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