  The goal of this paper is to develop numerical methods computing a few
smallest elasticity transmission eigenvalues, which are of practical importance
in inverse scattering theory. The problem is challenging since it is nonlinear,
non-self-adjoint, and of fourth order. We construct a nonlinear function whose
values are generalized eigenvalues of a series of self-adjoint fourth order
problems. The roots of the function are the transmission eigenvalues. Using an
$H^2$-conforming finite element for the self-adjoint fourth order eigenvalue
problems, we employ a secant method to compute the roots of the nonlinear
function. The convergence of the proposed method is proved. In addition, a
mixed finite element method is developed for the purpose of verification.
Numerical examples are presented to verify the theory and demonstrate the
effectiveness of the two methods.
