Massively parallel Schwarz methods for the high frequency Helmholtz equation

TL;DR

This work tackles solving the high-frequency Helmholtz equation by a massively parallel one-level overlapping Schwarz method with PML transmission (RAS-PML) on Cartesian subdomain coverings. A practical variant is developed where local subdomain problems include PML layers and impedance-type boundary conditions, with a partition of unity to assemble global updates; crucially, overlap and PML width can decrease with frequency as $\mathcal{O}(k^{-1}\log k)$ while maintaining convergence, enabling scalable parallelism. Numerical experiments in 2D demonstrate near-linear growth of runtime with frequency for a domain decomposed into $N=\mathcal{O}(k^2)$ subdomains and using $12$ points per wavelength, with RAS-PML-Imp offering the best robustness and impedance-based boundaries outperforming Dirichlet. The study shows that appropriate scaling of $\delta$ and $\kappa$ yields scalable performance, and extensions to variable wavespeed and 3D are planned for future work.

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.