Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
(2018) In Journal of Computational and Applied Mathematics 330. p.177-192- Abstract
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... (More)
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.
(Less)
- author
- Araujo-Cabarcas, Juan Carlos ; Engström, Christian LU and Jarlebring, Elias
- publishing date
- 2018-03-01
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Arnoldi's method, Dirichlet-to-Neumann map, Helmholtz problem, Matrix functions, Nonlinear eigenvalue problems, Scattering resonances
- in
- Journal of Computational and Applied Mathematics
- volume
- 330
- pages
- 16 pages
- publisher
- Elsevier
- external identifiers
-
- scopus:85029359070
- ISSN
- 0377-0427
- DOI
- 10.1016/j.cam.2017.08.012
- language
- English
- LU publication?
- no
- additional info
- Funding Information: We gratefully acknowledge the support of the Swedish Research Council under Grant No. 621-2012-3863 and 621-2013-4640 . J. Araújo also thanks the department of Mathematics at KTH Royal Institute of Technology very much for the kind hospitality and Giampaolo Mele for interesting discussions held during the visit. Publisher Copyright: © 2017 Elsevier B.V.
- id
- eac57d7f-3aed-443e-99dc-02f6c6e24b45
- date added to LUP
- 2023-03-24 11:06:57
- date last changed
- 2023-03-24 13:48:14
@article{eac57d7f-3aed-443e-99dc-02f6c6e24b45,
abstract = {{<p>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.</p>}},
author = {{Araujo-Cabarcas, Juan Carlos and Engström, Christian and Jarlebring, Elias}},
issn = {{0377-0427}},
keywords = {{Arnoldi's method; Dirichlet-to-Neumann map; Helmholtz problem; Matrix functions; Nonlinear eigenvalue problems; Scattering resonances}},
language = {{eng}},
month = {{03}},
pages = {{177--192}},
publisher = {{Elsevier}},
series = {{Journal of Computational and Applied Mathematics}},
title = {{Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map}},
url = {{http://dx.doi.org/10.1016/j.cam.2017.08.012}},
doi = {{10.1016/j.cam.2017.08.012}},
volume = {{330}},
year = {{2018}},
}