Recent Results on Domain Decomposition Preconditioning for the High-Frequency Helmholtz Equation Using Absorption

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 O(n 0.2 ) and times (on a serial machine) of order about O(n α ), with α ∈ [1.3, 1.4] for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size 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.

Stability and finite element error analysis for the Helmholtz equation with variable coefficients

“…There is considerable current interest in linear algebra problems arising from the discretization of heterogeneous wave problems (e.g. [20], [27]). The stability theory of the underlying PDE turned out to be key to rigorously understanding the performance of iterative methods in the homogeneous case (e.g.…”

Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

“…One advantage of DD over multigrid in this context is that "wave-based" components such as impedance or PML boundary conditions on the subdomains can more-easily be incorporated into DD preconditioners (see §1.5 below). The numerical experiments in [41] and [42] (following earlier experiments in [52] and [53]) show that additive Schwarz DD preconditioners for A can perform well for k ξ k 2 if impedance boundary conditions are used on the subdomain problems, instead of Dirichlet ones, and these experiments are backed up by analysis in [43] that shows that Property (ii) can be satisfied in some situations with |ξ| ∼ k 1+β for β small. 1.4.…”

Two-Level Preconditioners for the Helmholtz Equation

“…Let D j ∈ R nj ×nj be a diagonal matrix corresponding to a partition of unity in the sense that N sub i=1R T i R i = I, whereR j := D j R j . Then the one-level ORAS preconditioner (which is also the one-level ImpRAS of Graham et al [2017b]) reads…”

“…where * denotes the conjugate transpose, M −1 1,ε is the one-level preconditioner (3), Z is a rectangular matrix with full column rank, E = Z * A ε Z is the so-called coarse grid matrix, Ξ = ZE −1 Z * is the so-called coarse grid correction matrix. If P = Q = I this is an additive two-level preconditioner (which would be called two-level ImpRAS in Graham et al [2017b]). If P = I − A ε Ξ and Q = I − ΞA ε , this is a hybrid two-level preconditioner (ImpHRAS in Graham et al [2017b]), also called the Balancing Neumann Neumann (BNN) preconditioner.…”