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

TL;DR

The paper tackles numerical computation of scattering resonances for exterior acoustic problems with a sound-hard obstacle by reformulating resonances as eigenvalues of a holomorphic Fredholm operator $B(k)$ and truncating the unbounded domain with a Dirichlet-to-Neumann map $T(k)$. A finite element discretization using linear elements is analyzed via regular convergence theory for holomorphic Fredholm operator functions, combining DtN truncation error and FE discretization error to prove eigenvalue convergence. The nonlinear eigenproblem is solved efficiently with a parallel spectral indicator method (SIM), enabling simultaneous computation of multiple poles. Numerical experiments on a unit disk, unit square, and L-shaped domain validate convergence and demonstrate robustness and flexibility of the method, with exact disk poles recovered from analytic zeros of $H_m^{(1)'}(k)$ and domain-dependent convergence influenced by corner singularities. The approach provides a practical framework for computing scattering resonances in acoustics and can be extended to other scattering problems, including electromagnetic settings.

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.

Figures (4)

• Figure 1: The computed scattering poles and exact poles in $[0,4]\times[-4,0]$ for the unit disk.
• Figure 2: Real parts of the eigenfunctions associated with the three small scattering poles in Example 1.
• Figure 3: Real parts of the eigenfunctions associated with thethree small scattering poles in Example 2.
• Figure 4: Real parts of the eigenfunctions for the smallest three scattering poles of the L-shaped domain.

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• Definition 3.1
• Theorem 3.2
• Lemma 3.3
• Lemma 3.4
• proof