Can classical Schwarz methods for time-harmonic elastic waves converge?

Brunet, Romain and Dolean, Victorita and Gander, Martin J.; Haynes, Ronald and MacLachlan, Scott and Cai, Xiao-Chuan and Halpern, Laurence and Hyun Kim, Hyea and Klawonn, Axel and Widlund, Olof B, eds. (2020) Can classical Schwarz methods for time-harmonic elastic waves converge? In: Domain Decomposition Methods in Science and Engineering XXV. Springer-Verlag, Cham, Switzerland, pp. 425-432. ISBN 978-3-030-56750-7

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Abstract

We show that applying a classical Schwarz method to the time harmonic Navier equations, which are an important model for linear elasticity, leads in general to a divergent method for low to intermediate frequencies. This is even worse than for Helmholtz and time harmonic Maxwell's equations, where the classical Schwarz method is also not convergent, but low frequencies only stagnate, they do not diverge. We illustrate the divergent modes by numerical examples, and also show that when using the classical Schwarz method as a preconditioner for a Krylov method, convergence difficulties remain.

ORCID iDs

Brunet, Romain ORCID logoORCID: https://orcid.org/0000-0001-9761-9253, Dolean, Victorita ORCID logoORCID: https://orcid.org/0000-0002-5885-1903 and Gander, Martin J.; Haynes, Ronald, MacLachlan, Scott, Cai, Xiao-Chuan, Halpern, Laurence, Hyun Kim, Hyea, Klawonn, Axel and Widlund, Olof B