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

TL;DR

This work analyzes overlapping Schwarz methods with local Cartesian PMLs for the Helmholtz problem at high frequency and smooth wave speed. Building on prior semi-classical analysis, it shows that, under suitable k-independent overlaps and PML widths, the method achieves error decay faster than any power of k after a finite number of iterations, with the required number of subdomain intersections determined by geometric-optic rays. The paper extends this theory by experimentally testing regimes where overlap and PML width decrease as k increases, finding robustness for constant wavespeed even when the PML is about one wavelength wide and overlaps are minimal. The findings support the practicality of k-robust, parallelizable domain-decomposition solvers for high-frequency Helmholtz problems in realistic settings.

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.

Figures (1)

• Figure 1: RAS-PML/RMS-PML on Checkerboard: The rate of reduction of residual against $k$. Left: Reduction rate per $N_x+N_y -1$ iterations for RAS-PML; Right: Reduction rate per exhaustive sweep for RMS-PML

Theorems & Definitions (1)

• remark 1