Two-level preconditioners for the Helmholtz equation

TL;DR

This work evaluates two coarse-space definitions for two-level ORAS preconditioners applied to the Helmholtz equation $-\Delta u - k^2 u = f$ in 2D and 3D. It contrasts a grid-based coarse space on a coarse mesh with a DtN coarse space constructed from local Dirichlet-to-Neumann eigenproblems, within an absorption-augmented preconditioning framework where the target problem uses $\varepsilon=0$. The study shows that using absorption $\varepsilon_{\text{prec}}=k$ markedly improves convergence over $\varepsilon_{\text{prec}}=k^2$, and that the DtN coarse space reduces iterations relative to the grid coarse space for comparable coarse-space budgets, albeit at larger coarse-space sizes; with appropriate scaling of subdomain and coarse-mesh diameters $H_{\text{sub}} \sim k^{-\ alpha}$ and $H_{\text{coarse}} \sim k^{- alpha'}$ (and in 3D a balancing $\nalpha' = 3/2 - \nalpha$), the iteration counts grow only slowly with the wavenumber $k$ for large-scale problems (up to $>10^7$ unknowns). The results provide practical guidance on coarse-space design and parameter tuning for scalable Helmholtz solvers in 2D and 3D, with noted potential for extension to heterogeneous media.

Abstract

In this paper we compare numerically two different coarse space definitions for two-level domain decomposition preconditioners for the Helmholtz equation, both in two and three dimensions. While we solve the pure Helmholtz problem without absorption, the preconditioners are built from problems with absorption. In the first method, the coarse space is based on the discretization of the problem with absorption on a coarse mesh, with diameter constrained by the wavenumber. In the second method, the coarse space is built by solving local eigenproblems involving the Dirichlet-to-Neumann (DtN) operator.