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 the discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, the resulted discrete eigenvalues can be in a state of chaos, and consequently a large portion of them can be reliable and approximate the exact ones with high accuracy. Furthermore, we design an algorithm to select such a quasi-optimal penalty parameter. Finally, we provide numerical tests to demonstrate the performance of the proposed method.
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