  In this paper we make a further discussion on the finite elements
approximation for the Steklov eigenvalue problem on concave polygonal domain.
We make full use of the regularity estimate and the characteristic of edge
average interpolation operator of nonconforming Crouzeix-Raviart element, which
is different from the existing proof argument, and prove a new and optimal
error estimate in $\|\cdot\|_{0,\partial\Omega}$ for the eigenfunction of
linear conforming finite element and the nonconforming Crouzeix-Raviart
element, which is an improvement of the current results. Finally, we present
some numerical experiments to support the theoretical analysis.
