Core Concepts
The authors propose an efficient finite element DtN method to compute acoustic scattering resonances for sound hard obstacles, and provide a convergence analysis using the abstract approximation theory for holomorphic Fredholm operator functions.
Abstract
The key highlights and insights of the content are:
Scattering resonances are the poles of the meromorphic continuation of the scattering operator to the lower half-plane, and they have important applications in various fields. However, the numerical computation of scattering resonances faces several challenges.
The authors reformulate the scattering resonances as the eigenvalues of a holomorphic Fredholm operator function, and truncate the unbounded domain using the Dirichlet-to-Neumann (DtN) mapping, which is equivalent to the original problem and does not pollute the spectrum.
The linear Lagrange finite element method is used for discretization, and the convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic Fredholm operator functions.
The resulting nonlinear algebraic eigenvalue problem is solved efficiently using the recently developed parallel spectral indicator method (SIM), which can compute multiple poles simultaneously without requiring initial guesses.
Numerical examples for the unit disk and unit square obstacles are provided, demonstrating the effectiveness and convergence properties of the proposed method.
Stats
The authors provide the following key figures and metrics to support their analysis:
The exact scattering poles for the unit disk can be found analytically, and the computed poles on the finest mesh are consistent with the exact values.
The convergence orders of the computed scattering poles for the unit disk and unit square are approximately 2nd order.
The real parts of the eigenfunctions associated with the computed scattering poles are presented for both examples.