  We propose, analyze mathematically, and study numerically a novel approach
for the finite element approximation of the spectrum of second-order elliptic
operators. The main idea is to reduce the stiffness of the problem by
subtracting a least-squares penalty on the gradient jumps across the mesh
interfaces from the standard stiffness bilinear form. This penalty bilinear
form is similar to the known technique used to stabilize finite element
approximations in various contexts. The penalty term is designed to dampen the
high frequencies in the spectrum and so it is weighted here by a negative
coefficient. The resulting approximation technique is called softFEM since it
reduces the stiffness of the problem. The two key advantages of softFEM over
the standard Galerkin FEM are to improve the approximation of the eigenvalues
in the upper part of the discrete spectrum and to reduce the condition number
of the stiffness matrix. We derive a sharp upper bound on the softness
parameter weighting the stabilization bilinear form so as to maintain
coercivity for the softFEM bilinear form. Then we prove that softFEM delivers
the same optimal convergence rates as the standard Galerkin FEM approximation
for the eigenvalues and the eigenvectors. We next compare the discrete
eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed
analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a
sensible choice for the softness parameter. With this choice, the stiffness
reduction ratio scales linearly with the polynomial degree. Various numerical
experiments illustrate the benefits of using softFEM over Galerkin FEM.
