The Nonconforming Crouzeix-Raviart Element Approximation and Two-Grid Discretizations for the Elastic Eigenvalue Problem (Journal of Computational Mathematics, 2023)
In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321\u2013338(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 thenonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lam\u00e9 constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-griddiscretization schemes for the elastic eigenvalue problem, and analyze that when the meshsizes of coarse grid and fine grid satisfy some relationship, the resulting solutions canachieve the optimal accuracy. Numerical examples are provided to show the efficiency oftwo-grid schemes for the elastic eigenvalue problem.