On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

TL;DR

This paper investigates the use of Dirichlet-to-Neumann coarse spaces in domain decomposition methods for the Helmholtz problem, highlighting conditions for wave number independent convergence and robustness in heterogeneous settings.

Contribution

It introduces a novel threshold selection for eigenfunctions in the coarse space, improving convergence properties for Helmholtz problems.

Findings

Wave number independent convergence achieved in special cases.

Method converges quickly for homogeneous problems with large wave numbers.

Approach remains robust regardless of the number of subdomains.

Abstract

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.