On spurious solutions in finite element approximations of resonances in open systems
This work addresses the problem of spurious resonances in finite element simulations for researchers in computational acoustics and electromagnetics, but the results are limited to one-dimensional cases and incremental in nature.
The authors detect spurious solutions in finite element computations of resonances when using PML or DtN truncation in the pre-asymptotic regime, and propose a test based on the Lippmann-Schwinger equation and pseudospectrum to distinguish spurious from true eigenvalues.
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