  In this paper, using new correction to the Crouzeix-Raviart finite element
eigenvalue approximations, we obtain lower eigenvalue bounds for the Steklov
eigenvalue problem with variable coefficients on d-dimensional domains (d =
2,3). In addition, we prove that the corrected eigenvalues asymptotically
converge to the exact ones from below whether the eigenfunctions are singular
or smooth and whether the eigenvalues are large enough or not. Further, we
prove that the corrected eigenvalues still maintain the same convergence order
as that of uncorrected eigenvalues. Finally, numerical experiments validate our
theoretical results.
