New error estimates of linear triangle finite elements for the Steklov eigenvalue problem

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.