In this paper we present a mathematical and numerical analysis of an eigenvalue problem associated to the elasticity-Stokes equations stated in two and three dimensions. Both problems are related through the Herrmann pressure. Employing the Babu\v ska--Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, the finite element method is based on general inf-sup stables pairs for the Stokes system, such that, Taylor--Hood finite elements. By using a general approximation theory for compact operators, we obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Under mild assumptions, we have that these estimates hold with constants independent of the Lam\'e coefficient $\lambda$. In addition, we carry out the reliability and efficiency analysis of a residual-based a posteriori error estimator for the spectral problem. We report a series of numerical tests in order to assess the performance of the method and its behavior when the nearly incompressible case of elasticity is considered.

翻译：本文对与二维和三维弹性-斯托克斯方程相关联的特征值问题进行了数学与数值分析。这两个问题通过赫尔曼压力相关联。利用巴布斯卡-布雷齐理论，证明了连续和离散变分 formulations 的适定性。特别地，有限元方法基于斯托克斯系统的一般inf-sup稳定对，例如泰勒-胡德有限元。通过使用紧算子的通用逼近理论，我们获得了特征函数的最优阶误差估计以及特征值的双阶估计。在温和假设下，这些估计的常数与拉梅系数 $\lambda$ 无关。此外，我们还对基于残差的后验误差估计子进行了可靠性和效率分析。我们报告了一系列数值实验，以评估该方法的性能及其在近不可压缩弹性情况下的表现。