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

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.

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Cited by in corpus (10)

• Schwarz methods by domain truncation
• Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems
• Multidirectionnal sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems
• A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems
• GenEO coarse spaces for heterogeneous indefinite elliptic problems
• Inexact subdomain solves using deflated GMRES for Helmholtz problems
• Matrix-Free Parallel Scalable Multilevel Deflation Preconditioning for Heterogeneous Time-Harmonic Wave Problems
• Can Symmetric Positive Definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?
• Multipreconditioning with directional sweeping methods for high-frequency Helmholtz problems
• Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?