Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

TL;DR

This paper investigates the robustness of GenEO and DtN coarse spaces in a two-level additive Schwarz method for heterogeneous Helmholtz problems, demonstrating promising numerical results for large wave numbers and heterogeneity.

Contribution

The study introduces a GenEO-type coarse space for Helmholtz problems and compares its effectiveness with DtN coarse space, showing improved robustness and scalability.

Findings

Robust convergence with iteration counts independent of wave number

Good scalability demonstrated on 2D heterogeneous Helmholtz problems

GenEO coarse space compares favorably with DtN in numerical tests

Abstract

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.