Massively parallel Schwarz methods for the high frequency Helmholtz equation

TL;DR

This paper introduces a scalable parallel Schwarz method with PML conditions for high-frequency Helmholtz equations, demonstrating good convergence and scalability in 2D experiments as frequency increases.

Contribution

It presents a practical variant of the restricted additive Schwarz method with PML, allowing overlap and PML layer widths to decrease with frequency, ensuring efficiency at high frequencies.

Findings

Achieves (k^d) parallel scalability in experiments.

Maintains k iteration counts as frequency increases.

Demonstrates effective convergence in 2D Helmholtz problems.

Abstract

We investigate the parallel one-level overlapping Schwarz method for solving finite element discretization of high-frequency Helmholtz equations. The resulting linear systems are large, indefinite, ill-conditioned, and complex-valued. We present a practical variant of the restricted additive Schwarz method with Perfectly Matched Layer transmission conditions (RAS-PML), which was originally analyzed in a theoretical setting in {\tt arXiv:2404.02156}, with some numerical experiments given in {\tt arXiv:2408.16580}. In our algorithm, the width of the overlap and the additional PML layer on each subdomain is allowed to decrease with $\mathcal{O}(k^{-1} \log(k))$, as the frequency $k \rightarrow \infty$, and this is observed to ensure good convergence while avoiding excessive communication. In experiments, the proposed method achieves $\mathcal{O}(k^d)$ parallel scalability under Cartesian domain decomposition and exhibits $\mathcal{O}(k)$ iteration counts and convergence time for $d$-dimensional Helmholtz problems ($d = 2,3$) as $k$ increases. In this preliminary note we restrict to experiments on 2D problems with constant wave speed. Details, analysis and extensions to variable wavespeed and 3D will be given in future work.