Analysis of a finite element DtN method for scattering resonances of sound hard obstacles

TL;DR

This paper develops and analyzes a finite element DtN method for computing scattering resonances around sound hard obstacles, proving convergence and validating with numerical examples.

Contribution

It introduces a novel finite element DtN approach for scattering resonances and proves its convergence using abstract approximation theory.

Findings

Convergence of the discretized resonances is established.

Numerical methods successfully compute scattering resonances.

Validation confirms the effectiveness of the proposed approach.

Abstract

Scattering resonances have important applications in many areas of science and engineering. They are the replacement of discrete spectral data for problems on non-compact domains. In this paper, we consider the computation of scattering resonances defined on the exterior to a compact sound hard obstacle. The resonances are the eigenvalues of a holomorphic Fredholm operator function. We truncate the unbounded domain and impose the Dirichlet-to-Neumann (DtN) mapping. The problem is then discretized using the linear Lagrange element. Convergence of the resonances is proved using the abstract approximation theory for holomorphic Fredholm operator functions. The discretization leads to nonlinear algebraic eigenvalue problems, which are solved by the recently developed parallel spectral indicator methods. Numerical examples are presented for validation.