A Finite Element Based Fast Eigensolver for Three Dimensional Anisotropic Photonic Crystals

So-Hsiang Chou Bowling Green State University Tsung-Ming Huang National Taiwan Normal University Tiexiang Li Southeast University Jia-Wei Lin National Chiao Tung University Wen-Wei Lin National Chiao Tung University

Numerical Linear Algebra mathscidoc:1802.26001

Journal of Computational Physics, 386, 611-631, 2019.6
The standard Yee's scheme for the Maxwell eigenvalue problems places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor filed it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of the matrix is as large as $5,184,000$.
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@inproceedings{so-hsiang2019a,
  title={A Finite Element Based Fast Eigensolver for Three Dimensional Anisotropic Photonic Crystals},
  author={So-Hsiang Chou, Tsung-Ming Huang, Tiexiang Li, Jia-Wei Lin, and Wen-Wei Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180225230111689086934},
  booktitle={ Journal of Computational Physics},
  volume={386},
  pages={611-631},
  year={2019},
}
So-Hsiang Chou, Tsung-Ming Huang, Tiexiang Li, Jia-Wei Lin, and Wen-Wei Lin. A Finite Element Based Fast Eigensolver for Three Dimensional Anisotropic Photonic Crystals. 2019. Vol. 386. In Journal of Computational Physics. pp.611-631. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180225230111689086934.
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