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 u ? ? H3(?) H 3 ( ? ) , the consistency error is of order O(h2) O ( h 2 ) which is one order higher than its interpolation error O(h) O ( h ) ) and the interpolation postprocessing technique, the superclose and superconvergence results of order O(h2) O ( h 2 ) for the exact solution of eigenvector u u in broken H1 H 1 -norm are deduced on generalized rectangular meshes and rectangular meshes, respectively. Secondly, it is proved that the consistency error even can reach O(h4) O ( h 4 ) order for arbitrary quadrilateral meshes when u u ? ? H5(?) H 5 ( ? ) . 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) O ( h 4 ) for eigenvalue is derived. Finally, some numerical results are provided to verify the theoretical analysis.
eigenvalue problem modified quasi-wilson element superclose and superconvergence extrapolation