  The aim of the paper is to introduce a new systematic method that can produce
lower bounds for eigenvalues. The main idea is to use nonconforming finite
element methods. The general conclusion herein is that if local approximation
properties of nonconforming finite element spaces $V_h$ are better than global
continuity properties of $V_h$, corresponding methods will produce lower bounds
for eigenvalues. More precisely, under three conditions on continuity and
approximation properties of nonconforming finite element spaces we first show
abstract error estimates of approximate eigenvalues and eigenfunctions.
Subsequently, we propose one more condition and prove that it is sufficient to
guarantee nonconforming finite element methods to produce lower bounds for
eigenvalues of symmetric elliptic operators. As one application, we show that
this condition hold for most nonconforming elements in literature. As another
important application, this condition provides a guidance to modify known
nonconforming elements in literature and to propose new nonconforming elements.
In fact, we enrich locally the Crouzeix-Raviart element such that the new
element satisfies the condition; we propose a new nonconforming element for
second order elliptic operators and prove that it will yield lower bounds for
eigenvalues. Finally, we prove the saturation condition for most nonconforming
elements.
