Guaranteed Lower and Upper Bounds for Eigenvalues of Second Order Elliptic Operators in any Dimension

A new method is proposed to provide guaranteed lower bounds for eigenvalues of general second order elliptic operators in any dimension. This method employs a novel generalized Crouzeix–Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method. As a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based on generalized Crouzeix–Raviart element approximate eigenfunctions, an averaging interpolation from the generalized Crouzeix–Raviart element space to the conforming linear element space, and an usual Rayleigh–Ritz procedure. The ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix–Raviart element, explicitly computable constants for two interpolation operators. Numerical experiments demonstrate that the guaranteed lower bounds for eigenvalues in this paper are superior to those obtained by the Crouzeix–Raviart element.