A Penalized Crouzeix–Raviart Element Method for Second Order Elliptic Eigenvalue Problems

In this paper we propose a penalized Crouzeix–Raviart element method for eigenvalue problems of second order elliptic operators. The key idea is to add a penalty term to tune the local approximation property and the global continuity property of discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, some of the resulted discrete eigenvalues are upper bounds of exact ones, and the others are lower bounds, and consequently a large portion of them can be reliable and approximate eigenvalues with high accuracy. Furthermore, we design an algorithm to select a penalty parameter which meets the condition. Finally we provide numerical tests to demonstrate the performance of the proposed method.