Massively parallel Schwarz methods for the high frequency Helmholtz equation

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.

💡 Research Summary

The paper addresses the challenging task of solving high‑frequency Helmholtz equations, whose finite‑element discretizations lead to very large, indefinite, ill‑conditioned, complex‑valued linear systems. The authors develop a practical variant of the restricted additive Schwarz (RAS) method in which each subdomain is equipped with a perfectly matched layer (PML) and an impedance boundary condition. The key novelty lies in allowing both the overlap width (δ) and the additional PML thickness (κ) to shrink with the frequency k according to
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