  In this paper, we give a numerical analysis for the transmission eigenvalue
problem by the finite element method. A type of multilevel correction method is
proposed to solve the transmission eigenvalue problem. The multilevel
correction method can transform the transmission eigenvalue solving in the
finest finite element space to a sequence of linear problems and some
transmission eigenvalue solving in a very low dimensional spaces. Since the
main computational work is to solve the sequence of linear problems, the
multilevel correction method improves the overfull efficiency of the
transmission eigenvalue solving. Some numerical examples are provided to
validate the theoretical results and the efficiency of the proposed numerical
scheme.
