Virtual element method for the modified transmission eigenvalue problem in inverse scattering theory

In this paper, we present the virtual element approximation for the modified transmission eigenvalue problem. The eigenvalue problem arises in inverse scattering theory for artificial inhomogeneous media characterized as a metamaterial. By the shifting argument and spectral approximation theory of compact operators, we prove the correct spectral approximation of the virtual element scheme, and derive the optimal order error estimates for the eigenfunctions and the double order for the eigenvalues. Finally, we implement numerical experiments to illustrate behaviors of the standard and serendipity virtual element schemes, and the finite element scheme.