A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain

TL;DR

This paper introduces a novel pure source transfer domain decomposition method (PSTDDM) for efficiently solving Helmholtz equations in unbounded domains, improving scalability and convergence over previous methods.

Contribution

The paper develops a simplified PSTDDM that reduces subdomain problem size and enhances parallelism, with proven convergence for constant wave numbers and applicability as a preconditioner.

Findings

PSTDDM provides accurate approximations for Helmholtz problems.

The method is effective as a preconditioner in GMRES.

Convergence is proven for constant wave number cases.

Abstract

We propose a pure source transfer domain decomposition method (PSTDDM) for solving the truncated perfectly matched layer (PML) approximation in bounded domain of Helmholtz scattering problem. The method is a modification of the STDDM proposed by [Z. Chen and X. Xiang, SIAM J. Numer. Anal., 51 (2013), pp. 2331--2356]. After decomposing the domain into $N$ non-overlapping layers, the STDDM is composed of two series steps of sources transfers and wave expansions, where $N-1$ truncated PML problems on two adjacent layers and $N-2$ truncated half-space PML problems are solved successively. While the PSTDDM consists merely of two parallel source transfer steps in two opposite directions, and in each step $N-1$ truncated PML problems on two adjacent layers are solved successively. One benefit of such a modification is that the truncated PML problems on two adjacent layers can be further solved by the PSTDDM along directions parallel to the layers. And therefore, we obtain a block-wise PSTDDM on the decomposition composed of $N^2$ squares, which reduces the size of subdomain problems and is more suitable for large-scale problems. Convergences of both the layer-wise PSTDDM and the block-wise PSTDDM are proved for the case of constant wave number. Numerical examples are included to show that the PSTDDM gives good approximations to the discrete Helmholtz equations with constant wave numbers and can be used as an efficient preconditioner in the preconditioned GMRES method for solving the discrete Helmholtz equations with constant and heterogeneous wave numbers.