2016/06/30 by Juan Carlos Araujo-Cabarcas, Araujo-Cabarcas, Juan Carlos, Christian Engström +3
Computer Science · Engineering · Physics and Astronomy · #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1606.09547
openalex publication_date 2016/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an efficient procedure for computing resonances and resonant modes\nof Helmholtz problems posed in exterior domains. The problem is formulated as a\nnonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use\nof a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains.\nWe consider a variational formulation and show that the spectrum consists of\nisolated eigenvalues of finite multiplicity that only can accumulate at\ninfinity. The proposed method is based on a high order finite element\ndiscretization combined with a specialization of the Tensor Infinite Arnoldi\nmethod. Using Toeplitz matrices, we show how to specialize this method to our\nspecific structure. In particular we introduce a pole cancellation technique in\norder to increase the radius of convergence for computation of eigenvalues that\nlie close to the poles of the matrix-valued function. The solution scheme can\nbe applied to multiple resonators with a varying refractive index that is not\nnecessarily piecewise constant. We present two test cases to show stability,\nperformance and numerical accuracy of the method. In particular the use of a\nhigh order finite element discretization together with TIAR results in an\nefficient and reliable method to compute resonances.\n