Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique

Martin J. Gander; Hui Zhang

doi:https://doi.org/10.1007/978-3-030-95025-5_69

Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique

Martin J. Gander, Hui Zhang^*

^*Corresponding author for this work

Department of Applied Mathematics

University of Geneva

Research output: Chapter in Book or Report/Conference proceeding › Conference Proceeding › peer-review

Abstract

In 1990, Bruno Després published a short note [5] in Comptes rendus de l’Académie des sciences. Série 1, Mathématique. “The aim of this work is, after construction of a domain decomposition method adapted to the Helmholtz problem, to show its convergence.” The idea has been further developed in [12], [3], [9], [4] and [2] by employing radiation conditions with a special structure, subdomains without overlap and iterations in parallel or one-sweep. As of today, it seems the unique means by which Schwarz iterations (not Krylov-Schwarz) for the Helmholtz equation have been proved to converge in general geometry and variable media; otherwise, e.g., using PML ([1]) as boundary conditions requires the (sub)domain to be convex. This paper is to show that those algorithmic parameters are difficult to perturb even in a rectangle while maintaining convergent Schwarz iterations.

Translated title of the contribution	区域分解与亥姆霍兹方程：三十年后仍然独特
Original language	English
Title of host publication	Domain Decomposition Methods in Science and Engineering XXVI
Editors	Susanne C. Brenner, Axel Klawonn, Jinchao Xu, Eric Chung, Jun Zou, Felix Kwok
Place of Publication	Cham
Publisher	Springer Verlag
Pages	633-640
Number of pages	8
ISBN (Electronic)	978-3-030-95025-5
ISBN (Print)	978-3-030-95024-8
DOIs	https://doi.org/10.1007/978-3-030-95025-5_69
Publication status	Published - Mar 2023
Event	26th International Conference on Domain Decomposition Methods, 2020 - Virtual, Online Duration: 7 Dec 2020 → 12 Dec 2020

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	145
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	26th International Conference on Domain Decomposition Methods, 2020
City	Virtual, Online
Period	7/12/20 → 12/12/20

Access to Document

https://doi.org/10.1007/978-3-030-95025-5_69

Cite this

Gander, M. J., & Zhang, H. (2023). Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique. In S. C. Brenner, A. Klawonn, J. Xu, E. Chung, J. Zou, & F. Kwok (Eds.), Domain Decomposition Methods in Science and Engineering XXVI (pp. 633-640). (Lecture Notes in Computational Science and Engineering; Vol. 145). Springer Verlag. https://doi.org/10.1007/978-3-030-95025-5_69

Gander, Martin J. ; Zhang, Hui. / Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique. Domain Decomposition Methods in Science and Engineering XXVI . editor / Susanne C. Brenner ; Axel Klawonn ; Jinchao Xu ; Eric Chung ; Jun Zou ; Felix Kwok. Cham : Springer Verlag, 2023. pp. 633-640 (Lecture Notes in Computational Science and Engineering).

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abstract = "In 1990, Bruno Despr{\'e}s published a short note [5] in Comptes rendus de l{\textquoteright}Acad{\'e}mie des sciences. S{\'e}rie 1, Math{\'e}matique. “The aim of this work is, after construction of a domain decomposition method adapted to the Helmholtz problem, to show its convergence.” The idea has been further developed in [12], [3], [9], [4] and [2] by employing radiation conditions with a special structure, subdomains without overlap and iterations in parallel or one-sweep. As of today, it seems the unique means by which Schwarz iterations (not Krylov-Schwarz) for the Helmholtz equation have been proved to converge in general geometry and variable media; otherwise, e.g., using PML ([1]) as boundary conditions requires the (sub)domain to be convex. This paper is to show that those algorithmic parameters are difficult to perturb even in a rectangle while maintaining convergent Schwarz iterations.",

author = "Gander, {Martin J.} and Hui Zhang",

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Gander, MJ & Zhang, H 2023, Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique. in SC Brenner, A Klawonn, J Xu, E Chung, J Zou & F Kwok (eds), Domain Decomposition Methods in Science and Engineering XXVI . Lecture Notes in Computational Science and Engineering, vol. 145, Springer Verlag, Cham, pp. 633-640, 26th International Conference on Domain Decomposition Methods, 2020, Virtual, Online, 7/12/20. https://doi.org/10.1007/978-3-030-95025-5_69

Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique. / Gander, Martin J.; Zhang, Hui.
Domain Decomposition Methods in Science and Engineering XXVI . ed. / Susanne C. Brenner; Axel Klawonn; Jinchao Xu; Eric Chung; Jun Zou; Felix Kwok. Cham: Springer Verlag, 2023. p. 633-640 (Lecture Notes in Computational Science and Engineering; Vol. 145).

Research output: Chapter in Book or Report/Conference proceeding › Conference Proceeding › peer-review

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AU - Zhang, Hui

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N2 - In 1990, Bruno Després published a short note [5] in Comptes rendus de l’Académie des sciences. Série 1, Mathématique. “The aim of this work is, after construction of a domain decomposition method adapted to the Helmholtz problem, to show its convergence.” The idea has been further developed in [12], [3], [9], [4] and [2] by employing radiation conditions with a special structure, subdomains without overlap and iterations in parallel or one-sweep. As of today, it seems the unique means by which Schwarz iterations (not Krylov-Schwarz) for the Helmholtz equation have been proved to converge in general geometry and variable media; otherwise, e.g., using PML ([1]) as boundary conditions requires the (sub)domain to be convex. This paper is to show that those algorithmic parameters are difficult to perturb even in a rectangle while maintaining convergent Schwarz iterations.

AB - In 1990, Bruno Després published a short note [5] in Comptes rendus de l’Académie des sciences. Série 1, Mathématique. “The aim of this work is, after construction of a domain decomposition method adapted to the Helmholtz problem, to show its convergence.” The idea has been further developed in [12], [3], [9], [4] and [2] by employing radiation conditions with a special structure, subdomains without overlap and iterations in parallel or one-sweep. As of today, it seems the unique means by which Schwarz iterations (not Krylov-Schwarz) for the Helmholtz equation have been proved to converge in general geometry and variable media; otherwise, e.g., using PML ([1]) as boundary conditions requires the (sub)domain to be convex. This paper is to show that those algorithmic parameters are difficult to perturb even in a rectangle while maintaining convergent Schwarz iterations.

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T2 - 26th International Conference on Domain Decomposition Methods, 2020

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Gander MJ, Zhang H. Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique. In Brenner SC, Klawonn A, Xu J, Chung E, Zou J, Kwok F, editors, Domain Decomposition Methods in Science and Engineering XXVI . Cham: Springer Verlag. 2023. p. 633-640. (Lecture Notes in Computational Science and Engineering). doi: https://doi.org/10.1007/978-3-030-95025-5_69

Décomposition de Domaine et Problème de Helmholtz: Thirty Years After and Still Unique

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