On spurious solutions in finite element approximations of resonances in open systems

TL;DR

This paper investigates the occurrence of spurious solutions in finite element methods for computing resonances in open systems, proposing a new test based on the Lippmann-Schwinger equation to distinguish true resonances from artifacts.

Contribution

It introduces a novel test using the Lippmann-Schwinger equation and pseudospectrum analysis to identify genuine resonances in finite element approximations.

Findings

The test effectively distinguishes true eigenvalues from spurious ones.

Spurious solutions are prevalent in the pre-asymptotic regime with domain truncation.

Numerical simulations validate the effectiveness of the proposed method.

Abstract

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann-Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases. Keywords: scattering resonances, Lippmann-Schwinger equation, nonlinear eigenvalue problems, acoustic resonator, dielectric resonator, Bragg resonator