An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation

TL;DR

This paper introduces a scalable additive overlapping domain decomposition method for efficiently solving high-frequency Helmholtz equations, with proven exactness in certain cases and demonstrated high performance in numerical experiments.

Contribution

It presents a novel additive overlapping domain decomposition method that is highly scalable and theoretically exact for the Helmholtz equation in specific settings.

Findings

Method is highly scalable and parallelizable.

Theoretical proof of exactness in constant medium case.

Numerical experiments show excellent performance.

Abstract

In this paper, we propose and analyze an additive domain decomposition method (DDM) for solving the high-frequency Helmholtz equation with the Sommerfeld radiation condition. In the proposed method, the computational domain is partitioned into structured subdomains along all spatial directions, and each subdomain contains an overlapping region for source transferring. At each iteration all subdomain PML problems are solved completely in parallel, then all horizontal, vertical and corner directional residuals on each subdomain are passed to its corresponding neighbor subdomains as the source for the next iteration. This DDM method is highly scalable in nature and theoretically shown to produce the exact solution for the PML problem defined in ${\mathbb{R}}^2$ in the constant medium case. A slightly modified version of the method for bounded truncated domains is also developed for its use in practice and an error estimate is rigorously proved. Various numerical experiments in two and three dimensions are conducted on the supercomputer "Tianhe-2 Cluster" to verify the theoretical results and demonstrate excellent performance of the proposed method as an iterative solver or a preconditioner.