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

TL;DR

This paper establishes sharp bounds on impedance-to-impedance maps for the Helmholtz equation at high frequencies, providing insights into the convergence and robustness of overlapping domain decomposition methods.

Contribution

It introduces new sharp bounds on impedance-to-impedance maps for Helmholtz at high wavenumber, advancing understanding of domain decomposition convergence.

Findings

Parallel Schwarz method is power contractive for two subdomains with large overlap, independent of wavenumber.

Composite impedance maps behave poorly with many subdomains, but show robustness for certain data classes.

Results provide theoretical support for observed numerical robustness of domain decomposition methods.

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].