Algorithm 922

Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues. Transmission eigenvalue problem has important applications in inverse scattering. Since the problem is non-self-adjoint, the computation of transmission eigenvalues needs special treatment. Based on a fourth-order reformulation of the transmission eigenvalue problem, a mixed finite element method is applied. The method has two major advantages: 1) the formulation leads to a generalized eigenvalue problem naturally without the need to invert a related linear system, and 2) the nonphysical zero transmission eigenvalue, which has an infinitely dimensional eigenspace, is eliminated. To solve the resulting non-Hermitian eigenvalue problem, an iterative algorithm using restarted Arnoldi method is proposed. To make the computation efficient, the search interval is decided using a Faber-Krahn type inequality for transmission eignevalues and the interval is updated at each iteration. The algorithm is implemented using Matlab. The code can be easily used in the qualitative methods in inverse scattering and modified to compute transmission eigenvalues for other models such as elasticity problem.

This software is also peer reviewed by journal TOMS.


References in zbMATH (referenced in 33 articles )

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  1. Meng, Jian; Wang, Gang; Mei, Liquan: A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem (2021)
  2. Ji, Xia; Li, Peijun; Sun, Jiguang: Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method (2020)
  3. Ren, Shixian; Tan, Ting; An, Jing: An efficient spectral-Galerkin approximation based on dimension reduction scheme for transmission eigenvalues in polar geometries (2020)
  4. Yang, Yidu; Han, Jiayu; Bi, Hai; Li, Hao; Zhang, Yu: Mixed methods for the elastic transmission eigenvalue problem (2020)
  5. Yang, Yidu; Zhang, Yu; Bi, Hai: A type of adaptive (C^0) non-conforming finite element method for the Helmholtz transmission eigenvalue problem (2020)
  6. Bi, Hai; Han, Jiayu; Yang, Yidu: Local and parallel finite element algorithms for the transmission eigenvalue problem (2019)
  7. Boujlida, H.; Haddar, H.; Khenissi, M.: The asymptotic of transmission eigenvalues for a domain with a thin coating (2018)
  8. Han, Jiayu: Nonconforming elements of class (L^2) for Helmholtz transmission eigenvalue problems (2018)
  9. Li, Hao; Yang, Yidu: An adaptive (C^0)IPG method for the Helmholtz transmission eigenvalue problem (2018)
  10. Li, Tiexiang; Huang, Tsung-Ming; Lin, Wen-Wei; Wang, Jenn-Nan: On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method (2018)
  11. Wang, Shixi; Bi, Hai; Zhang, Yu; Yang, Yidu: A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues (2018)
  12. Xi, Yingxia; Ji, Xia; Geng, Hongrui: A C(^0)IP method of transmission eigenvalues for elastic waves (2018)
  13. Li, Tiexiang; Huang, Tsung-Ming; Lin, Wen-Wei; Wang, Jenn-Nan: An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues (2017)
  14. Xie, Hehu; Wu, Xinming: A multilevel correction method for interior transmission eigenvalue problem (2017)
  15. Yang, Yidu; Bi, Hai; Li, Hao; Han, Jiayu: A (C^0 \mathrmIPG) method and its error estimates for the Helmholtz transmission eigenvalue problem (2017)
  16. An, Jing: A Legendre-Galerkin spectral approximation and estimation of the index of refraction for transmission eigenvalues (2016)
  17. Han, Jiayu; Yang, Yidu: An adaptive finite element method for the transmission eigenvalue problem (2016)
  18. Huang, Ruihao; Struthers, Allan A.; Sun, Jiguang; Zhang, Ruming: Recursive integral method for transmission eigenvalues (2016)
  19. Yang, Yidu; Han, Jiayu; Bi, Hai: Non-conforming finite element methods for transmission eigenvalue problem (2016)
  20. Zeng, Fang; Sun, JiGuang; Xu, LiWei: A spectral projection method for transmission eigenvalues (2016)

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