  We propose an efficient numerical method for a non-selfadjoint Steklov
eigenvalue problem. The Lagrange finite element is used for discretization. The
convergence is proved using the spectral perturbation theory for compact
operators. The non-sefadjointness of the problem leads to non-Hermitian matrix
eigenvalue problem. Due to the existence of complex eigenvalues and lack of a
priori spectral information, we propose a modified version of the recently
developed spectral indicator method to compute (complex) eigenvalues in a given
region on the complex plane. In particular, to reduce computational cost, the
problem is transformed into a much smaller matrix eigenvalue problem involving
the unknowns only on the boundary of the domain. Numerical examples are
presented to validate the effectiveness of the proposed method.
