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On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems

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Abstract

The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmetric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

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Notes

  1. Here \(\mathcal {O}\) denotes the “big O”.

  2. Since we are only interested in finding the positive real eigenvalues of the QEP (10), we restrict our discussion to the case \(\tau \ge 0\).

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Acknowledgments

The first author is supported by NSFC (No. 11471074) and the Fundamental Research Funds for the Central Universities. The fourth author is supported by NSFC (No. 91330109). The second and third authors would like to acknowledge the Grant support from the Ministry of Science and Technology (W.-Q. Huang: MOST 103-2811-M-009-029; W.-W. Lin: MOST 103-2628-M-009-005) and the National Center for Theoretical Sciences in Taiwan, and they would also like to thank the ST Yau Center at the National Chiao Tung University for the support.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Tiexiang Li & Jijun Liu

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wei-Qiang Huang & Wen-Wei Lin

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  1. Tiexiang Li
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Correspondence to Wei-Qiang Huang.

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Li, T., Huang, WQ., Lin, WW. et al. On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems. J Sci Comput 64, 83–108 (2015). https://doi.org/10.1007/s10915-014-9923-0

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  • DOI: https://doi.org/10.1007/s10915-014-9923-0

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