  In this paper, we present a multi-level mixed element scheme for the
Helmholtz transmission eigenvalue problem on polygonal domains that are not
necessarily able to be covered by rectangle grids. We first construct an
equivalent linear mixed formulation of the transmission eigenvalue problem and
then discretize it with Lagrangian finite elements of low regularities. The
proposed scheme admits a natural nested discretization, based on which we
construct a multi-level scheme. Optimal convergence rate and optimal com-
putational cost can be obtained with the scheme.
