Spurious-mode-free finite element method for scattering resonances in transmission problems

TL;DR

This paper introduces a novel finite element method that accurately computes scattering resonances in transmission problems without spurious modes, using a DtN map and spectrum indicator for eigenvalue calculation.

Contribution

The paper presents a spurious-mode-free finite element approach for scattering resonances, with proven convergence and practical effectiveness demonstrated through extensive numerical examples.

Findings

Achieves optimal order convergence in resonance computation.

Effectively eliminates spurious modes in numerical solutions.

Results align with and extend existing theoretical insights.

Abstract

Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from spurious modes. In this paper, we propose a spurious-mode-free method for computing scattering resonances in transmission problems. The unbounded domain is truncated using a Dirichlet-to-Neumann (DtN) map. The resonances are formulated as eigenvalues of a holomorphic Fredholm operator function, which is discretized by the finite element method. The spectrum indicator method is then used to compute the eigenvalues of the nonlinear matrix eigenvalue problems. We establish optimal order convergence and present extensive examples that demonstrate the effectiveness of the proposed method. The results are consistent with existing theoretical findings in the literature and offer new insights that may inform further theoretical developments.