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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:330

Article

Araujo-Cabarcas, Juan Carlos¹  Engstrom, Christian¹  Jarlebring, Elias² 

[1] Umea Univ, Dept Math & Math Stat, MIT Huset, Umea, Sweden

[2] Royal Inst Technol KTH, Dept Math, SeRC Swedish E Sci Res Ctr, Stockholm, Sweden

关键词: Nonlinear eigenvalue problems;    Helmholtz problem;    Scattering resonances;    Dirichlet-to-Neumann map;    Arnoldi's method;    Matrix functions;   

DOI  :  10.1016/j.cam.2017.08.012

来源: Elsevier

【 摘 要 】

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-toNeumann 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. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free