A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

Royer, Anthony; Geuzaine, Christophe; Béchet, Eric; Modave, Axel

doi:10.1016/j.cma.2022.115006

Article (Scientific journals)

A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

Royer, Anthony; Geuzaine, Christophe; Béchet, Eric et al.

2022 • In Computer Methods in Applied Mechanics and Engineering, 395, p. 115006

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https://hdl.handle.net/2268/292644

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10.1016/j.cma.2022.115006

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Keywords :

Cross-points; Domain decomposition; Finite elements; Helmholtz equation; Perfectly matched layer; Transmission condition; Cross point; Dirichlet-to-Neumann map; Domain decompositions; Domain-decomposition methods; Helmholtz's equations; Layer transmission; Non-overlapping domain decomposition methods; Perfectly Matched Layer; Subdomain; Transmission conditions; Computational Mechanics; Mechanics of Materials; Mechanical Engineering; Physics and Astronomy (all); Computer Science Applications; General Physics and Astronomy

Abstract :

[en] It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which can lead to slow convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care. In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.

Disciplines :

Electrical & electronics engineering
Mathematics

Author, co-author :

Royer, Anthony ; Université de Liège - ULiège > Montefiore Institute of Electrical Engineering and Computer Science

Geuzaine, Christophe ; Université de Liège - ULiège > Montefiore Institute of Electrical Engineering and Computer Science

Béchet, Eric ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)

Modave, Axel ; POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, Palaiseau, France

Language :

English

Title :

A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation

Publication date :

15 May 2022

Journal title :

Computer Methods in Applied Mechanics and Engineering

ISSN :

0045-7825

eISSN :

1879-2138

Publisher :

Elsevier B.V.

Volume :

395

Pages :

115006

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0045782522002420?httpAccept=text/xml

Funding text :

This research was funded in part through the ARC grant for Concerted Research Actions (ARC WAVES 15/19-03), financed by the Wallonia-Brussels Federation of Belgium .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 .

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