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

Romain Brunet; Victorita Dolean; Martin J. Gander

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

Romain Brunet, Victorita Dolean, Martin J. Gander

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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.

Original language	English
Title of host publication	Domain Decomposition Methods in Science and Engineering XXV
Editors	Ronald Haynes, Scott MacLachlan, Xiao-Chuan Cai, Laurence Halpern, Hyea Hyun Kim, Axel Klawonn, Olof B Widlund
Place of Publication	Cham, Switzerland
Publisher	Springer-Verlag
Pages	425-432
Number of pages	8
ISBN (Electronic)	978-3-030-56750-7
ISBN (Print)	978-3-030-56749-1
Publication status	Published - 25 Oct 2020

Keywords

Schwarz method
mathematical modeling
elastic media

Cite this

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

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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. ",

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Can classical Schwarz methods for time-harmonic elastic waves converge? / Brunet, Romain; Dolean, Victorita; Gander, Martin J.
Domain Decomposition Methods in Science and Engineering XXV. ed. / Ronald Haynes; Scott MacLachlan; Xiao-Chuan Cai; Laurence Halpern; Hyea Hyun Kim; Axel Klawonn; Olof B Widlund. Cham, Switzerland: Springer-Verlag, 2020. p. 425-432.