  In this paper, using the linearization technique we write the Helmholtz
transmission eigenvalue problem as an equivalent nonselfadjoint linear
eigenvalue problem whose left-hand side term is a selfadjoint, continuous and
coercive sesquilinear form. To solve the resulting nonselfadjoint eigenvalue
problem, we give an $H^{2}$ conforming finite element discretization and
establish a two grid discretization scheme. We present a complete error
analysis for both discretization schemes, and theoretical analysis and
numerical experiments show that the methods presented in this paper can
efficiently compute real and complex transmission eigenvalues.
