Radial complex scaling for anisotropic scalar resonance problems

TL;DR

This paper introduces a radial complex scaling method for anisotropic scalar resonance problems, addressing issues with traditional Cartesian scaling and demonstrating its theoretical and computational effectiveness.

Contribution

It proves the Fredholm property of the radial scaling operator and shows convergence of numerical approximations, rehabilitating complex scaling for anisotropic media.

Findings

Radial scaling yields correct radiation conditions in anisotropic media.

The associated operator is proven to be Fredholm.

Numerical methods converge to the true solution.

Abstract

The complex scaling/perfectly matched layer method is a widely spread technique to simulate wave propagation problems in open domains. The method is very popular, because its implementation is very easy and does not require the knowledge of a fundamental solution. However, for anisotropic media the method may yield an unphysical radiation condition and lead to erroneous and unstable results. In this article we argue that a radial scaling (opposed to a cartesian scaling) does not suffer from this drawback and produces the desired radiation condition. This result is of great importance as it rehabilitates the application of the complex scaling method for anisotropic media. To present further details we consider the radial complex scaling method for scalar anisotropic resonance problems. We prove that the associated operator is Fredholm and show the convergence of approximations generated by simulateneous domain truncation and finite element discretization. We present computational studies to undergird our theoretical results.