Non-iterative domain decomposition for the Helmholtz equation using the method of difference potentials

TL;DR

This paper introduces a non-iterative domain decomposition method for the Helmholtz equation using the Method of Difference Potentials, enabling efficient and robust solutions even with complex boundary conditions and large wavenumber jumps.

Contribution

The paper presents a novel non-iterative domain decomposition approach based on the Method of Difference Potentials for the Helmholtz equation, handling complex boundary conditions and high wavenumber contrasts.

Findings

Method is insensitive to interior cross-points.

Handles mixed boundary conditions effectively.

Robust against large jumps in wavenumber.

Abstract

We use the Method of Difference Potentials (MDP) to solve a non-overlapping domain decomposition formulation of the Helmholtz equation. The MDP reduces the Helmholtz equation on each subdomain to a Calderon's boundary equation with projection on its boundary. The unknowns for the Calderon's equation are the Dirichlet and Neumann data. Coupling between neighboring subdomains is rendered by applying their respective Calderon's equations to the same data at the common interface. Solutions on individual subdomains are computed concurrently using a straightforward direct solver. We provide numerical examples demonstrating that our method is insensitive to interior cross-points and mixed boundary conditions, as well as large jumps in the wavenumber for transmission problems, which are known to be problematic for many other Domain Decomposition Methods.