The Nonconforming Finite Element Methods for Two Transmission Eigenvalue Problems in Inverse Scattering

In this paper we study the nonconforming Crouzeix-Raviart element and the enriched Crouzeix-Raviart element methods firstly for the transmission eigenvalue problem of inhomogeneous media and the modified transmission eigenvalue problem in inverse scattering. Using $\mathbb{T}$-coercivity method, we prove the convergence and the a priori error estimate of approximate eigenpair for the transmission eigenvalue problem, and based on the obtained results we prove the a priori error estimate for the modified transmission eigenvalue problem by the $\mathbb{T}$-coercivity method and Gårding inequality, and further prove that the discrete eigenvalues for the problem with metamaterial background approximate the exact eigenvalue from above. We also carry out numerical experiments to validate the theoretical findings and the efficiency of the proposed methods.