Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

Cavity problem; Domain decomposition method; Helmholtz equation; Optimized Schwarz method; Cavity problems; Domain-decomposition methods; Helmholtz problems; Helmholtz's equations; Optimized Schwarz methods; Overlapping Schwarz; Propagating waves; Rectangular cavity; Schwarz method; Transmission operators; Modeling and Simulation; Computational Theory and Mathematics; Computational Mathematics; Mathematics - Numerical Analysis; Computer Science - Numerical Analysis

Abstract :

[en] In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.

Disciplines :

Mathematics

Author, co-author :

Marsic, Nicolas ; Technische Universität Darmstadt, Institute for Accelerator Science and Electromagnetic Fields (TEMF), Darmstadt, Germany

Geuzaine, Christophe ; Université de Liège - ULiège > Département d'électricité, électronique et informatique (Institut Montefiore) > Applied and Computational Electromagnetics (ACE)

De Gersem, Herbert; Technische Universität Darmstadt, Institute for Accelerator Science and Electromagnetic Fields (TEMF), Darmstadt, Germany

Language :

English

Title :

Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

Publication date :

15 May 2023

Journal title :

Computers and Mathematics with Applications

ISSN :

0898-1221

eISSN :

1873-7668

Publisher :

Elsevier Ltd

Volume :

138

Pages :

37 - 64

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0898122123000780?httpAccept=text/xml

Funding text :

This research project has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 445906998 . The work of Nicolas Marsic is also supported by the Graduate School CE within the Centre for Computational Engineering at the Technische Universität Darmstadt . Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region . The authors would like to express their gratitude to Mr. Anthony Royer for his help with the GmshFEM and GmshDDM frameworks. In addition, the authors are grateful to Ms. Heike Koch, Mr. Achim Wagner, Mr. Dragos Munteanu, Mr. Christian Schmitt, Dr. Wolfgang F.O. Müller and Dr. David Colignon for the administrative and technical support. Finally, the authors would like to thank the anonymous Reviewers, whose comments improved significantly the quality of this work.

Commentary :

37 pages, 23 figures. Changes with respect to the previous version: i) one more reference (original GMRES paper) and ii) fixing more typos. This version is published in Computers & Mathematics with Applications

Available on ORBi :

since 29 November 2024

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