  In this paper, we extend the work of Brenner and Sung [Math. Comp. 59,
321--338 (1992)] and present a regularity estimate for the elastic equations in
concave domains. Based on the regularity estimate we prove that the constants
in the error estimates of the nonconforming Crouzeix-Raviart element
approximations for the elastic equations/eigenvalue problem are independent of
the Lame constant, which means the nonconforming Crouzeix-Raviart element
approximations are locking-free. We also establish two kinds of two-grid
discretization schemes for the elastic eigenvalue problem and analyze that when
the mesh sizes of the coarse grid and fine grid satisfy some relationship, the
resulting solutions can achieve optimal accuracy. Numerical examples are
provided to show the efficiency of two-grid schemes for the elastic eigenvalue
problem.
