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Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0001-9443-8772
2018 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed) Published
Abstract [en]

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. 

Place, publisher, year, edition, pages
Elsevier, 2018. Vol. 330, p. 177-192
Keyword [en]
Arnoldi's method, Dirichlet-to-Neumann map, Helmholtz problem, Matrix functions, Nonlinear eigenvalue problems, Scattering resonances, Finite element method, Matrix algebra, Numerical methods, Poles, Refractive index, Resonance, Switching systems, Arnoldi's methods, Helmholtz problems, Nonlinear eigenvalue problem, Scattering resonance, Eigenvalues and eigenfunctions
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-216793DOI: 10.1016/j.cam.2017.08.012ISI: 000415783000014Scopus ID: 2-s2.0-85029359070OAI: oai:DiVA.org:kth-216793DiVA, id: diva2:1156703
Funder
Swedish Research Council, 621-2012-3863 621-2013-4640
Note

QC 20171205

Available from: 2017-11-14 Created: 2017-11-14 Last updated: 2017-12-05Bibliographically approved

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Jarlebring, Elias

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