Nonconforming finite element analysis for Poisson eigenvalue problem

The main aim of this paper is to study a nonconforming quadrilateral finite element (named modified quasi-Wilson element) approximation to Poisson eigenvalue problem. Firstly, by employing a special property of this element (when u∈H3(Ω), the consistency error is of order O(h2) which is one order higher than its interpolation error O(h)) and the interpolation postprocessing technique, the superclose and superconvergence results of order O(h2) for the exact solution of eigenvector u in broken H1-norm are deduced on generalized rectangular meshes and rectangular meshes, respectively. Secondly, it is proved that the consistency error even can reach O(h4) order for arbitrary quadrilateral meshes when u∈H5(Ω). This is a new astonishing feature which has never been discovered. Subsequently, based on the above characteristic and some asymptotic expansions of the conforming bilinear finite element, the extrapolation solution of order O(h4) for eigenvalue is derived. Finally, some numerical results are provided to verify the theoretical analysis.