Optimized Schwarz Methods for Heterogeneous Helmholtz and Maxwell’s Equations

Dolean, Victorita; Gander, Martin J.; Veneros, Erwin; Zhang, Hui

doi:10.1007/978-3-319-52389-7_13

Cited by 3 publications

(1 citation statement)

References 22 publications

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“…Starting with Caroline Japhet [61], a huge development began following the paradigm of approximating the optimal non-local operators by parametrized local ones and then optimizing the parameters with respect to the rate of convergence, typically on the continuous level, often the PDE-level, using Fourier analysis for the case of two subdomains sharing a common face (often two half-spaces). For a comprehensive survey on optimized Schwarz methods see [45,50], for the case of wave propagation see in particular [31,32,33,46,48,1].…”

Section: Domain Decomposition (Dd) Methodsmentioning

confidence: 99%

A unified theory of non-overlapping Robin-Schwarz methods -- continuous and discrete, including cross points

Pechstein¹

2022

Preprint

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Non-overlapping Schwarz methods with generalized Robin transmission conditions were originally introduced by B. Després for time-harmonic wave propagation problems and have largely developed over the past thirty years. The aim of the paper is to provide both a review of the available formulations and methods as well as a consistent theory applicable to more general cases than studied until to date. An abstract variational framework is provided reformulating the original problem by the well-known form involving a scattering operator and an interface exchange operator, and the equivalence between the formulations is discussed thoroughly. The framework applies to a series of wave propagation problems throughout the de Rham complex, such as the scalar Helmholtz equation, Maxwell's equations, a dual formulation of the Helmholtz equation in H(div), as well as any conforming finite element discretization thereof, and it applies also to coercive problems. Three convergence results are shown. The first one (using compactness) and the second one (based on absorbtion) generalize Després' early findings and apply as well to the FETI-2LM formulation (a discrete method introduced by de La Bourdonnaye, Farhat, Macedo, Magoulés, and Roux). The third result, oriented on the work by Collino, Ghanemi, and Joly, establishes a convergence rate and covers cases with cross points, while not requiring any regularity of the solution. The key ingredient is a global interface exchange operator, proposed originally by X. Claeys and further developed by Claeys and Parolin, here worked out in full generality. The third type of convergence theory is applicable at the discrete level as well, where the exchange operator is allowed to be even local. The resulting scheme can be viewed as a generalization of the 2-Lagrange-multiplier method introduced by S. Loisel, and connections are drawn to another technique proposed by Gander and Santugini.

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