 
Summary: ASYMPTOTIC LOWER BOUNDS FOR EIGENVALUES BY
NONCONFORMING FINITE ELEMENT METHODS
MAR´IA G. ARMENTANO AND RICARDO G. DUR´AN
Abstract. We analyze the approximation obtained for the eigenvalues of the Laplace operator by
the nonconforming piecewise linear finite element of CrouzeixRaviart . For singular eigenfunctions,
as those arising in non convex polygons, we prove that the eigenvalues obtained with this method
give lower bounds of the exact eigenvalues when the mesh size is small enough.
Key words. Finite elements, eigenvalue problems, nonconforming methods.
AMS subject classifications. 65N25,65N30
1. Introduction. For second order elliptic problems it is known that the eigen
values computed using the standard conforming finite element method are always
above the exact ones. Indeed this can be proved using the minimummaximum char
acterization of the eigenvalues (see for example [4]). Therefore, it is an interesting
problem to find methods which give lower bounds of the eigenvalues. However, as far
as we know, only few results in this direction have been obtained and mainly for finite
difference methods. Forsythe proved that the eigenvalue approximation obtained by
standard five points finite differences is below the eigenvalue of the continuous prob
lem, when the meshsize is small enough, for some particular domains and smooth
enough eigenfunctions [8], [9] . Since that finite difference method coincides with the
standard piecewise linear finite elements with mass lumping on uniform meshes, one
