  In this paper, a new method is proposed to produce guaranteed lower bounds
for eigenvalues of general second order elliptic operators in any dimension.
Unlike most methods in the literature, the proposed method only needs to solve
one discrete eigenvalue problem but not involves any base or intermediate
eigenvalue problems, and does not need any a priori information concerning
exact eigenvalues either. Moreover, it just assumes basic regularity of exact
eigenfunctions. This method is defined by 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 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. Numerics are provided to demonstrate
the theoretical results.
