The Lower/Upper Bound Property of Approximate Eigenvalues by Nonconforming Finite Element Methods for Elliptic Operators

Abstract

This paper is a complement of the work (Hu et al. in arXiv:1112.1145v1[math.NA], 2011), where a general theory is proposed to analyze the lower bound property of discrete eigenvalues of elliptic operators by nonconforming finite element methods. One main purpose of this paper is to propose a novel approach to analyze the lower bound property of discrete eigenvalues produced by the Crouzeix---Raviart element when exact eigenfunctions are smooth. In particular, under some conditions on the triangular mesh, it is proved that the Crouzeix---Raviart element method of the Laplace operator yields eigenvalues below exact ones. Such a theoretical result explains most of numerical results in the literature and also partially answers the problem of Boffi (Acta Numerica 1---120, 2010). This approach can be applied to the Crouzeix---Raviart element of the Stokes eigenvalue problem and the Morley element of the buckling eigenvalue problem of a plate. As a second main purpose, a new identity of the error of eigenvalues is introduced to study the upper bound property of eigenvalues by nonconforming finite element methods, which is successfully used to explain why eigenvalues produced by the rotated $$Q_1$$ Q 1 element of second order elliptic operators (when eigenfunctions are smooth), the Adini element (when eigenfunctions are singular) and the new Zienkiewicz-type element of fourth order elliptic operators, are above exact ones.