[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
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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