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The Finite Element Method for the Elastic Transmission Eigenvalue Problem with Different Elastic Tensors

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Abstract

The elastic transmission eigenvalue problem, arising from the inverse scattering theory, plays a critical role in the qualitative reconstruction methods for elastic media. In this paper, we discuss the finite element method for solving the elastic transmission eigenvalue problem with different elastic tensors and different mass densities. The problem is neither self-adjoint nor elliptic at any frequency. By the \({\mathbb {T}}\)-coercivity we prove that the spectral of the problem is the spectral of a compact operator, and by the \({\mathbb {T}}\)-coercivity and Babuška–Osborn spectral approximation theory we derive a priori error estimate of the discrete eigenpairs. Numerical experiments show that the method is easy to implement and can efficiently compute elastic transmission eigenvalues.

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Acknowledgements

The authors cordially thank the editor and the referees for their valuable comments and suggestions which lead to the improvement of this paper.

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Projects supported by the National Natural Science Foundation of China (Grant Nos. 12261024, 11561014).

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  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

    Yidu Yang, Shixi Wang & Hai Bi

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  1. Yidu Yang
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Projects supported by the National Natural Science Foundation of China (Grant Nos. 12261024, 11561014).

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Yang, Y., Wang, S. & Bi, H. The Finite Element Method for the Elastic Transmission Eigenvalue Problem with Different Elastic Tensors. J Sci Comput 93, 65 (2022). https://doi.org/10.1007/s10915-022-02030-3

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