Two-level preconditioners for the Helmholtz equation

TL;DR

This paper compares two coarse space strategies for two-level domain decomposition preconditioners applied to the Helmholtz equation, focusing on their effectiveness in 2D and 3D without absorption.

Contribution

It introduces and numerically compares two novel coarse space constructions for Helmholtz preconditioners, one based on absorption discretization and the other on eigenproblem solutions.

Findings

Both methods improve solver convergence.

The eigenproblem-based coarse space shows better scalability.

Numerical results validate the effectiveness of the proposed approaches.

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.