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Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map. (English) Zbl 1376.65142

Summary: 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 (TIAR). 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.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35N30 Overdetermined initial-boundary value problems for PDEs and systems of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Software:

NLEIGS; NLEVP; deal.ii; PETSc
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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