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This paper is conceed with 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 CrouzeixRaviart element,and prove a new and optimal error estimate in ｜｜· ｜｜0,(e)Ω for the eigenfunction of linear conforming finite element and the nonconforming Crouzeix-Raviart element.Finally,we present some numerical results to support the theoretical analysis.