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arXiv · 2211.14659

Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition

David Lafontaine·Euan A. Spence
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Abstract

We prove sharp bounds on certain impedance-to-impedance maps (and their compositions) for the Helmholtz equation with large wavenumber (i.e., at high-frequency) using semiclassical defect measures. The paper [GGGLS] (Gong-Gander-Graham-Lafontaine-Spence, 2022) recently showed that the behaviour of these impedance-to-impedance maps (and their compositions) dictates the convergence of the parallel overlapping Schwarz domain-decomposition method with impedance boundary conditions on the subdomain boundaries. For a model decomposition with two subdomains and sufficiently-large overlap, the results of this paper combined with those in [GGGLS] show that the parallel Schwarz method is power contractive, independent of the wavenumber. For strip-type decompositions with many subdomains, the results of this paper show that the composite impedance-to-impedance maps, in general, behave "badly" with respect to the wavenumber; nevertheless, by proving results about the composite maps applied to a restricted class of data, we give insight into the wavenumber-robustness of the parallel Schwarz method observed in the numerical experiments in [GGGLS].

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David Lafontaine, Euan A. Spence. 2024-02-01. Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition. https://doi.org/10.2140/paa.2023.5.927

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