A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues

In this paper, for the Helmholtz transmission eigenvalue problem, we propose a two-grid discretization scheme of non-conforming finite elements. With this scheme, the solution of the transmission eigenvalue problem on a fine grid πh is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid πH and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid πh. We prove the resulting solution still maintains an asymptotically optimal accuracy, and we report some numerical examples in two dimension and three dimension on the modified-Zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues.