Spurious-mode-free finite element method for scattering resonances in transmission problems
For researchers in wave phenomena and numerical analysis, this method provides a reliable tool for computing scattering resonances without spurious modes, though it is an incremental improvement over existing finite element approaches.
The paper proposes a spurious-mode-free finite element method for computing scattering resonances in transmission problems, achieving optimal order convergence and eliminating spurious modes that plague existing methods.
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.