  In this paper, we use the non-conforming Crouzeix-Raviart element method to
solve a Stekloff eigenvalue problem arising in inverse scattering. The weak
formulation corresponding to this problem is non-selfadjoint and does not
satisfy $H^{1}$-elliptic condition,and its Crouzeix-Raviart element
discretization does not meet the Strang lemma condition. We use the standard
duality techniques to prove an extension of Strang lemma. And we prove the
convergence and error estimate of discrete eigenvalues and eigenfunctions using
the spectral perturbation theory for compact operators. Finally, we present
some numerical examples not only on uniform meshes but also in an adaptive
refined meshes to show that the Crouzeix-Raviart method is efficient for
computing real and complex eigenvalues as expected.
