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：本文的目的是分析一种用于线性弹性本征值问题的混合方法，它通过分段的$(k+1)$，$k$和$(k+1)$次多项式函数($k\geq1$)来数值近似应力、位移和旋转。应力的数值特征函数是对称的。通过数值位移的离散$H^1$稳定性，我们证明了在适当的正则性假设下，该方法与精确位移的本征空间的$L^2$正交投影之间的误差是$O(h^{k+2})$。因此，通过后处理，我们比传统方法得到了更好的对精确位移的本征空间的近似。我们还证明了应力数值逼近的自由锁定特性。我们介绍了一种混合方法来将混合方法简化为一种凝聚本征问题，并使用数值位移的离散$H^1$稳定性证明了非线性本征问题的初始近似（独立于弹性算子的逆）的误差为$O(h^2)$，而传统的inf-sup条件只能得到$O(h)$的逼近误差。最后，我们报告了一些数值实验结果。