Non-conforming Finite Element And Mixed Finite Element Two-grid Discretization Schemes For Transmission Eigenvalue Problem

The transmission eigenvalue problem is a quadratic eigenvalue problem arising in inverse scattering theory for an inhomogeneous medium.Transmission eigenvalues can be used to obtain estimates for the material properties of the scattering object,and have theoretical importance in the uniqueness and reconstruction in inverse scattering theory.In this paper,based on the non-conforming finite element formulation and the mixed finite element formulations,we propose non-conforming finite element and mixed finite element two-grid discretization schemes for the Helmholtz transmission eigenvalue problem.With these schemes,the solution of the transmission eigenvalue problem on a fine grid ?_H is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid ?_H.For the two-grid discretization scheme of non-conforming finite elements,we prove the resulting solution still maintains an asymptotically optimal accuracy,and we report some numerical experiments on the modified-Zienkiewicz element in both two dimension and three dimension to validate the efficiency of our approach.For the two-grid discretization scheme of mixed finite elements,the several numerical examples demonstrate the effectiveness of those schemes.