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

TL;DR

This paper develops locking-free nonconforming Crouzeix-Raviart element methods and two-grid discretizations for elastic eigenvalue problems, providing error estimates independent of Lame constants and demonstrating optimal accuracy through numerical experiments.

Contribution

It extends regularity estimates to concave domains and proves error bounds for nonconforming elements that are independent of material parameters, introducing efficient two-grid schemes.

Findings

Error estimates are independent of Lame constants.

Two-grid schemes achieve optimal accuracy.

Numerical examples confirm efficiency of proposed methods.

Abstract

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.