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Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
Umeå university, Sweden.
Umeå university, Sweden.
KTH Royal Instute of Technology, Sweden.
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
Amsterdam: Elsevier, 2018. Vol. 330, p. 177-192
Keywords [en]
Nonlinear eigenvalue problems, Helmholtz problem, Scattering resonances, Dirichlet-to-Neumann map, Arnoldi's method, Matrix functions
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-81922DOI: 10.1016/j.cam.2017.08.012ISI: 000415783000014OAI: oai:DiVA.org:lnu-81922DiVA, id: diva2:1304786
Available from: 2019-04-13 Created: 2019-04-13 Last updated: 2019-05-07Bibliographically approved

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Araujo-Cabarcas, Juan CarlosEngström, Christian

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CiteExportLink to record
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  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
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More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
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Output format
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