A comparison of coarse spaces for Helmholtz problems in the high frequency regime

TL;DR

This paper compares three coarse space strategies for high-frequency Helmholtz problems, analyzing their performance and suitability through numerical experiments to identify the most effective approaches for challenging wave propagation applications.

Contribution

It provides a comparative analysis of grid, DtN, and GenEO coarse spaces for two-level domain decomposition methods in high-frequency Helmholtz problems, highlighting their advantages and limitations.

Findings

GenEO performs well in heterogeneous media.

DtN coarse space is effective for smooth problems.

Grid coarse space is simple but less robust in complex settings.

Abstract

Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending on the problem and particular numerical setting.