A rapidly converging domain decomposition method for the Helmholtz equation

TL;DR

This paper introduces a new domain decomposition method for the Helmholtz equation that achieves rapid convergence with near-linear computational cost, effectively handling wave reflections and scaling well with problem size.

Contribution

The paper presents a novel domain decomposition approach using PML-based transmission conditions that significantly reduces iteration counts and computational complexity for Helmholtz problems.

Findings

Near-linear computational cost achieved in large-scale examples

Iteration numbers are nearly independent of problem size

Effective handling of wave reflections at subdomain interfaces

Abstract

A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D Helmholtz equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. We focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes, in combination with a certain ordering for solving the subdomain problems and a GMRES outer iteration. When combined with multifrontal methods, the solver has near-linear cost in examples, due to very small iteration numbers that are essentially independent of problem size and number of subdomains. It is to our knowledge only the second method with this property next to the moving PML sweeping method.