Numerical assessment of PML transmission conditions in a domain decomposition method for the Helmholtz equation

TL;DR

This paper develops an efficient Helmholtz equation solver using PML transmission conditions within an optimized domain decomposition framework, demonstrating improved convergence and accuracy in numerical simulations.

Contribution

It introduces a novel PML-based transmission condition in an ORAS domain decomposition method for Helmholtz problems, enhancing solver performance.

Findings

PML conditions improve convergence rates.

Numerical results show higher accuracy with PMLs.

Method effective in 2D and 3D domains.

Abstract

The convergence rate of domain decomposition methods (DDMs) strongly depends on the transmission condition at the interfaces between subdomains. Thus, an important aspect in improving the efficiency of such solvers is careful design of appropriate transmission conditions. In this work, we will develop an efficient solver for Helmholtz equations based on perfectly matched layers (PMLs) as transmission conditions at the interfaces within an optimised restricted additive Schwarz (ORAS) domain decomposition preconditioner, in both two and three dimensional domains. We perform a series of numerical simulations on a model problem and will assess the convergence rate and accuracy of our solutions compared to the situation where impedance boundary conditions are used.