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

TL;DR

This paper introduces an efficient numerical method combining high-order finite elements and a specialized Arnoldi approach to accurately compute resonances in Helmholtz problems with unbounded domains, applicable to multiple resonators.

Contribution

It develops a novel approach using a Dirichlet-to-Neumann map and a pole cancellation technique within a Tensor Infinite Arnoldi framework for resonance computation.

Findings

Method achieves high accuracy in resonance calculations.

Demonstrates stability and efficiency on test cases.

Applicable to complex refractive index configurations.

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.