Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map

Abstract

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method. Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

Shape optimization for the strong directional scattering of dielectric nanorods

“…This, from the accuracy of the solutions and performance of the solver. 18,19 These conclusions can also be drawn for the scattering problems treated in this work. We emphasize this point in Section 3.5.1 where formula (62) gives a pre asymptotic estimate for controlling the pollution error.…”

“…The implementation of the truncated DtN operator used in this work is based on the implementation by Araújo et al 18…”

“…The advantage of high-order FEM for wave scattering problems with smooth interfaces is greatly experienced when comparing dispersive pollution error of low-order versus high-order FE. Pre asymptotic error estimation predicts that increasing the polynomial degree p is more efficient than decreasing the mesh size h. The works by Araújo et al 18,19 present results for scattering resonance computations by comparing the resulting errors obtained by refining p and h, respectively. Particularly, h refinements consists on fixing p and decreasing h uniformly over the mesh, and p refinements consists of increasing p uniformly and keeping h fixed.…”

“…Convergence of FE methods for Helmholtz problems with a truncated DtN map as boundary condition was proven by Melenk. 27 Numerical studies for scattering 26 and resonance problems 18 suggest that the truncation rule | | = l > ⌈ a⌉ results in accurate FE computations.…”

Shape optimization for the strong routing of light in periodic diffraction gratings

“…In practice, a truncated operator R l m is used instead of the full series (14). Similarly as for Dirichlet-to-Newmann formulations [53,54], the accuracy of the FE computation depends critically on the number of terms used in the truncation of the series.…”