A Weak Galerkin Finite Element Method Can Compute Both Upper and Lower Eigenvalue Bounds

In this paper, we observe an interesting phenomenon for a weak Galerkin (WG) approximation of eigenvalue problems, that is, we may obtain good approximations for exact eigenvalues from below or above, only through adjusting the global penalty parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence order with lower expenses. Some new techniques are developed to analyze upper and lower bound properties of eigenvalues. Numerical results supporting our theory are given.