A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems

Tsung-Ming Huang National Taiwan Normal University Wei-Qiang Huang National Chiao Tung University Wen-Wei Lin National Chiao Tung University

Numerical Analysis and Scientific Computing mathscidoc:1702.25082

SIAM J. Sci. Comput., 37, A2403-A2423, 2015
We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell’s transmission eigenvalue problem. The discretized governing equations by the Nedelec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form XY^{−1}Z, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient method (LOBPCG), to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG. Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenvalues.
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@inproceedings{tsung-ming2015a,
  title={A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems},
  author={Tsung-Ming Huang, Wei-Qiang Huang, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170223081920253301508},
  booktitle={SIAM J. Sci. Comput.},
  volume={37},
  pages={A2403-A2423},
  year={2015},
}
Tsung-Ming Huang, Wei-Qiang Huang, and Wen-Wei Lin. A robust numerical algorithm for computing Maxwell's transmission eigenvalue problems. 2015. Vol. 37. In SIAM J. Sci. Comput.. pp.A2403-A2423. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170223081920253301508.
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