Recent Results on Domain Decomposition Preconditioning for the High-frequency Helmholtz Equation using Absorption
Provides efficient multilevel solvers for high-frequency Helmholtz problems, relevant to computational acoustics and electromagnetics.
This paper reviews domain decomposition preconditioners for Helmholtz problems using absorption, achieving iteration counts of O(n^0.2) and solve times of O(n^{1.3-1.4}) for 2D problems up to 50 wavelengths, both for pollution-free and fixed-grid-per-wavelength discretizations.
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 $\mathcal{O}(n^α)$, { with $α\in [1.3,1.4]$} 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.