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.

翻译：在本文中,我们扩展了Brenner和Sung[Math. Comp. 59, 321-338 (1992)]的工作,并对凝固域中的弹性方程提出了定期性估计。根据定期性估计,我们证明,不兼容的Crouzeix-Raviart 元素对弹性方程/基因价值问题的误差估计值的常数与Lame 常数无关,这意味着不兼容的Crouzix-Raviart 元素近似是无锁的。我们还为弹性电子元值问题制定了两种双电网分解计划,并分析了当粗粗网格和精细网格的网格大小满足某些关系时,由此产生的解决办法能够达到最佳的准确性。提供了数字实例,以显示弹性电子值问题的双电网计划的效率。