This paper studies the family of interior penalty discontinuous Galerkin methods for solving the Herrmann formulation of the linear elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lam\'e coefficient norm within the framework of non-compact operators theory, we prove convergence of both continuous and discrete eigenvalue problems as the mesh size approaches zero, independently of the Lam\'e constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.

翻译：本文研究了一类内惩罚间断伽辽金方法在含异质介质中求解赫尔曼公式线性弹性特征值问题的应用。通过在非紧算子理论框架下采用加权拉梅系数范数，我们证明了当网格尺寸趋近于零时，连续与离散特征值问题均收敛，且收敛性独立于拉梅常数。此外，我们进行了后验误差分析，并提出了一种可靠且高效的估计子。数值实验验证了理论结果。