Recent Results on Domain Decomposition Preconditioning for the High-frequency Helmholtz Equation using Absorption

TL;DR

This paper reviews recent advances in domain decomposition preconditioners for high-frequency Helmholtz equations with absorption, highlighting their analysis, implementation, and performance in large-scale 2D problems.

Contribution

It introduces new preconditioners based on absorption for Helmholtz problems, with theoretical analysis and practical experiments demonstrating their efficiency.

Findings

Iteration counts grow as approximately n^{0.2}

Computational times scale as approximately n^{1.3 to 1.4}

Effective for problems up to about 50 wavelengths

Abstract

In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about $50$ wavelengths are presented. The experiments show iteration counts of order about $\mathcal{O}(n^{0.2})$ and times (on a serial machine) of order about $\mathcal{O}(n^{\alpha})$, { with $\alpha \in [1.3,1.4]$} for solving systems of dimension $n$. This holds both in the pollution-free case corresponding to meshes with grid size $\mathcal{O}(k^{-3/2})$ (as the wavenumber $k$ increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.