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arXiv · 1802.03687

Computation of Transmission Eigenvalues for Elastic Waves

Xia Ji·Peijun Li·Jiguang Sun
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Abstract

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$-conforming finite element for the self-adjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.

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Xia Ji, Peijun Li, Jiguang Sun. 2018-02-11. Computation of Transmission Eigenvalues for Elastic Waves. https://arxiv.org/abs/1802.03687

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