The Nonconforming Finite Element Methods for Two Transmission Eigenvalue Problems in Inverse Scattering (Advances in Applied Mathematics and Mechanics, 2025)
In this paper we study the nonconforming Crouzeix-Raviart element andthe enriched Crouzeix-Raviart element methods firstly for the transmission eigenvalueproblem of inhomogeneous media and the modified transmission eigenvalue problemin inverse scattering. Using $\mathbb{T}$-coercivity method, we prove the convergence and thea 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 themodified 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 outnumerical experiments to validate the theoretical findings and the efficiency of theproposed methods.