Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM

In this paper, we analyze some approximation properties of a nonconforming piecewise linear finite element with integral degrees of freedom. A nonconforming finite element method (FEM) is applied to second-order eigenvalue problems (EVPs). We prove that the eigenvalues computed by means of this element are smaller than the exact ones if the mesh size is small enough. The case when an EVP is defined on a nonconvex domain is considered.
A superconvergent rate is established to a second-order elliptic problem by the introduction of nonstandard interpolated elements based on the integral type linear element. A simple postprocessing method applied to second-order EVPs is also proposed and analyzed.
Finally, computational aspects are discussed and numerical examples are presented.