In this paper, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi-Douglas-Marini mixed finite element is also established. Numerical results supporting our theory are given.

翻译：本文针对特征值问题的可杂交间断伽辽金（HDG）方法观察到一个有趣现象：使用相同的有限元方法，仅通过微调稳定化参数即可同时获得特征值的上下界。基于这一观察，我们设计了一种高精度特征值计算算法，能以较低计算成本实现更高收敛速率。同时，我们证明特定类型的HDG方法仅能提供上界。作为副产品，本文还建立了Brezzi-Douglas-Marini混合有限元的渐近上界特性。数值结果验证了理论分析。