Schwarz methods with PMLs for Helmholtz problems: fast convergence at high frequency

TL;DR

This paper extends previous theoretical results on Schwarz methods with PMLs for Helmholtz problems by experimentally demonstrating robustness at high frequencies even with minimal overlap and PML width proportional to wavelength.

Contribution

The paper provides experimental evidence that Schwarz methods with PMLs remain effective at high frequencies when overlap and PML width decrease with wavenumber, extending prior theoretical work.

Findings

Methods remain robust at high wavenumber with minimal overlap.

PML width of one wavelength maintains effectiveness.

Experimental validation supports theoretical predictions.

Abstract

We discuss parallel (additive) and sequential (multiplicative) variants of overlapping Schwarz methods for the Helmholtz equation in $\mathbb{R}^d$, with large real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The domain-decomposition subdomains are overlapping hyperrectangles with Cartesian PMLs at their boundaries. In a recent paper ({\tt arXiv:2404.02156}), the current authors proved (for both variants) that, after a specified number of iterations -- depending on the behaviour of the geometric-optic rays -- the error is smooth and smaller than any negative power of the wavenumber $k$. For the parallel method, the specified number of iterations is less than the maximum number of subdomains, counted with their multiplicity, that a geometric-optic ray can intersect. The theory, which is given at the continuous level and makes essential use of semi-classical analysis, assumes that the overlaps of the subdomains and the widths of the PMLs are all independent of the wavenumber. In this paper we extend the results of {\tt arXiv:2404.02156} by experimentally studying the behaviour of the methods in the practically important case when both the overlap and the PML width decrease as the wavenumber increases. We find that (at least for constant wavespeed), the methods remain robust to increasing $k$, even for miminal overlap, when the PML is one wavelength wide.