The purpose of this paper is to analyze a mixed method for linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise $(k+1)$, $k$ and $(k+1)$-th degree polynomial functions ($k\geq 1$), respectively. The numerical eigenfunction of stress is symmetric. By the discrete $H^1$-stability of numerical displacement, we prove an $O(h^{k+2})$ approximation to the $L^{2}$-orthogonal projection of the eigenspace of exact displacement for the eigenvalue problem, with proper regularity assumption. Thus via postprocessing, we obtain a better approximation to the eigenspace of exact displacement for the eigenproblem than conventional methods. We also prove that numerical approximation to the eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce the mixed method to a condensed eigenproblem and prove an $O(h^2)$ initial approximation (independent of the inverse of the elasticity operator) of the eigenvalue for the nonlinear eigenproblem by using the discrete $H^1$-stability of numerical displacement, while only an $O(h)$ approximation can be obtained if we use the traditional inf-sup condition. Finally, we report some numerical experiments.

翻译：本文旨在分析一种用于线性弹性特征值问题的混合方法，该方法分别通过分片 $(k+1)$ 次、$k$ 次和 $(k+1)$ 次多项式函数（$k\geq 1$）对数值应力、位移和旋转进行近似。数值应力特征函数是对称的。借助数值位移的离散 $H^1$ 稳定性，在适当的正则性假设下，我们证明了特征值问题中精确位移特征空间的 $L^2$ 正交投影具有 $O(h^{k+2})$ 的逼近阶。因此，通过后处理，我们获得了比传统方法更优的精确位移特征空间逼近。我们还证明了应力数值特征函数的近似关于泊松比是无锁的。我们引入一种混合化方法，将混合方法简化为一个凝聚特征值问题，并利用数值位移的离散 $H^1$ 稳定性，证明非线性特征值问题中特征值具有 $O(h^2)$ 的初始逼近（与弹性算子逆无关），而若使用传统的 inf-sup 条件仅能得到 $O(h)$ 的逼近。最后，我们报告了一些数值实验。