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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 inconcave domains. Based on the regularity estimate we prove that the constantsin the error estimates of the nonconforming Crouzeix-Raviart elementapproximations for the elastic equations/eigenvalue problem are independent ofthe Lame constant, which means the nonconforming Crouzeix-Raviart elementapproximations are locking-free. We also establish two kinds of two-griddiscretization schemes for the elastic eigenvalue problem and analyze that whenthe mesh sizes of the coarse grid and fine grid satisfy some relationship, theresulting solutions can achieve optimal accuracy. Numerical examples areprovided to show the efficiency of two-grid schemes for the elastic eigenvalueproblem.

The nonconforming Crouzeix-Raviart element approximation and two-grid discretizations for the elastic eigenvalue problem