Why classical schwarz methods applied to certain hyperbolic systems converge even without overlap

Dolean, Victorita and Gander, Martin J.; (2008) Why classical schwarz methods applied to certain hyperbolic systems converge even without overlap. In: Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computational Science and Engineering, 60 . Springer, AUT, pp. 467-475. ISBN 9783540751984 (https://doi.org/10.1007/978-3-540-75199-1_59)

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Abstract

Overlap is essential for the classical Schwarz method to be convergent when solving elliptic problems. Over the last decade, it was however observed that when solving systems of hyperbolic partial differential equations, the classical Schwarz method can be convergent even without overlap. We show that the classical Schwarz method without overlap applied to the Cauchy-Riemann equations which represent the discretization in time of such a system, is equivalent to an optimized Schwarz method for a related elliptic problem, and thus must be convergent, since optimized Schwarz methods are well known to be convergent without overlap.

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