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Fourier-Galerkin method for scattering poles of sound soft obstacles

Yunyun Ma, Jiguang Sun

TL;DR

This paper tackles the numerical computation of scattering poles (complex resonances) for sound-soft obstacles in two dimensions. The poles are the zeros of boundary integral operators, specifically the single-layer operator $\mathcal{S}(\kappa)$ or the operator $\mathcal{I}+\mathcal{D}(\kappa)$, in the region $\{\operatorname{Im}(\kappa)<0\}$. A Fourier-Galerkin discretization is developed and shown to converge regularly, with eigenvalue error estimates derived from the abstract theory for holomorphic Fredholm operator functions (Karma 1996). For the second formulation, an auxiliary operator $\mathcal{H}(\kappa)$ is constructed that shares eigenvalues with $\mathcal{I}+\mathcal{D}(\kappa)$ and is holomorphic Fredholm, enabling a unified convergence theory. Numerical experiments on multiple obstacle geometries validate accuracy, demonstrate rapid convergence, and confirm that the boundary-only discretization yields no spurious modes, aided by a parallel spectral indicator method with contour integrals.

Abstract

The computation of scattering poles for a sound-soft obstacle is investigated. These poles correspond to the eigenvalues of two boundary integral operators. We construct novel decompositions of these operators and show that they are Fredholm. Then a Fourier-Galerkin method is proposed for discretization. By establishing the regular convergence of the discrete operators, an error estimate is established using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. We give details of the numerical implementation. Several examples are presented to validate the theory and demonstrate the effectiveness of the proposed method.

Fourier-Galerkin method for scattering poles of sound soft obstacles

TL;DR

This paper tackles the numerical computation of scattering poles (complex resonances) for sound-soft obstacles in two dimensions. The poles are the zeros of boundary integral operators, specifically the single-layer operator or the operator , in the region . A Fourier-Galerkin discretization is developed and shown to converge regularly, with eigenvalue error estimates derived from the abstract theory for holomorphic Fredholm operator functions (Karma 1996). For the second formulation, an auxiliary operator is constructed that shares eigenvalues with and is holomorphic Fredholm, enabling a unified convergence theory. Numerical experiments on multiple obstacle geometries validate accuracy, demonstrate rapid convergence, and confirm that the boundary-only discretization yields no spurious modes, aided by a parallel spectral indicator method with contour integrals.

Abstract

The computation of scattering poles for a sound-soft obstacle is investigated. These poles correspond to the eigenvalues of two boundary integral operators. We construct novel decompositions of these operators and show that they are Fredholm. Then a Fourier-Galerkin method is proposed for discretization. By establishing the regular convergence of the discrete operators, an error estimate is established using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. We give details of the numerical implementation. Several examples are presented to validate the theory and demonstrate the effectiveness of the proposed method.

Paper Structure

This paper contains 5 sections, 8 theorems, 74 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Integral Equations for Scattering Poles
  3. Fourier Galerkin Method
  4. Numerical Implementation and Examples
  5. Conclusions and Discussions

Key Result

Theorem 2.1

Let $\kappa \in \mathbb C$ and $\Gamma$ be of $C^{p+3,1}$ with $p\geq 0$.

Figures (4)

  • Figure 1: $\log{{\rm RIM}_m(\kappa)}$ for Example \ref{['exa:disk']} : Left: $\mathcal{S}_n(\kappa)$. Right: $\boldsymbol{I}_{2n+1}+\boldsymbol{D}_n(\kappa)$
  • Figure 2: absolute error for Example \ref{['exa:disk']} : Left: $\kappa_1$. Right: $\kappa_2$
  • Figure 3: $\log{{\rm RIM}_m(\kappa)}$ for Example \ref{['exa:peanut']} : Left: $\mathcal{S}_n(\kappa)$. Right: $\boldsymbol{I}_{2n+1}+\boldsymbol{D}_n(\kappa)$
  • Figure 4: $\log{{\rm RIM}_m(\kappa)}$ for Example \ref{['exa:acorn']} : Left: $\mathcal{S}_n(\kappa)$. Right: $\boldsymbol{I}_{2n+1}+\boldsymbol{D}_n(\kappa)$

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.2
  • proof
  • ...and 10 more