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Edge element DtN method for electromagnetic scattering poles of perfectly conducting obstacles

Bo Gong, Takumi Sato, Jiguang Sun, Xinming Wu

Abstract

Meromorphic continuation of the scattering operator leads to scattering poles (resonances) in the complex plane. Despite their significance, numerical investigation of scattering poles remains limited. In this paper, we propose and analyze a numerical method to compute electromagnetic poles of perfectly conducting obstacles. The unbounded domain for the scattering problem is truncated using the DtN mapping and the poles are shown to be the eigenvalues of a holomorphic Fredholm operator function related to Maxwell's equations. Edge elements are used for discretization. The convergence is proved using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. The proposed finite element DtN approach is free of non-physical poles. A spectral indicator method is then employed to compute the resulting nonlinear matrix eigenvalue problem. Numerical examples are presented to demonstrate the effectiveness of the method.

Edge element DtN method for electromagnetic scattering poles of perfectly conducting obstacles

Abstract

Meromorphic continuation of the scattering operator leads to scattering poles (resonances) in the complex plane. Despite their significance, numerical investigation of scattering poles remains limited. In this paper, we propose and analyze a numerical method to compute electromagnetic poles of perfectly conducting obstacles. The unbounded domain for the scattering problem is truncated using the DtN mapping and the poles are shown to be the eigenvalues of a holomorphic Fredholm operator function related to Maxwell's equations. Edge elements are used for discretization. The convergence is proved using the abstract approximation theory for eigenvalue problems of holomorphic Fredholm operator functions. The proposed finite element DtN approach is free of non-physical poles. A spectral indicator method is then employed to compute the resulting nonlinear matrix eigenvalue problem. Numerical examples are presented to demonstrate the effectiveness of the method.
Paper Structure (13 sections, 14 theorems, 118 equations, 2 figures, 4 tables)

This paper contains 13 sections, 14 theorems, 118 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Scattering poles of PEC obstacles
  3. Transparent boundary condition
  4. Weak formulation and operator equation
  5. Truncation of the Calderón operator
  6. Edge Element Method
  7. Abstract approximation theory
  8. Convergence result for the eigenvalues
  9. Numerical Examples
  10. Unit Ball
  11. Unit Cube
  12. L-shaped Domain
  13. Cavity

Key Result

Lemma 2.1

For $\kappa\in\Lambda$, the sesquilinear form $a_\kappa^{(1)}$ satisfies the following inf-sup condition where $C$ is a constant depending only on the constants $\alpha_1, \alpha_2$.

Figures (2)

  • Figure 4.1: Poles for the unit ball (zeros of $d_n(\kappa)$).
  • Figure 4.2: Computed resonances for the unit ball.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 17 more