Juan Carlos Araujo-Cabarcas: Finite element computation of scattering frequencies of the Helmholtz resonator

In the study of wave propagation in open domains with dielectric scatterers, the localization of energy peaks is closely related to the scattering resonances. Computing resonances of the transverse electromagnetic scattering problem in open domains requires setting up an outgoing condition. In particular we compare two well known methods to do so: Dirichlet to neumann maps (DtN) and the perfectly matched layer (PML). The use of the DtN results in a fully nonlinear eigenvalue problem where the spectral parameter is argument of trascendental functions in dimensions 2 and 3. While the PML method gives the advantage that the resulting rational eigenvalue problem is linear for non-frequency-dependent materials. However, the PML method introduces spectral pollution and reduces the area where we look for resonances. We discretize in space by using high order finite element methods and observe that in both cases spurious eigenvalues appear due to the discretization. We notice that the spurious eigenvalues and the polynomial order in use are closely related to the number of spurious eigenvalues and we present a way of reducing/eliminating them.