Nonconforming finite element analysis for Poisson eigenvalue problem

Abstract

a b s t r a c tThe 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 ∈ H 3 (Ω), the consistency error is of order O(h 2 ) which is one order higher than its interpolation error O(h)) and the interpolation postprocessing technique, the superclose and superconvergence results of order O(h 2 ) for the exact solution of eigenvector u in broken 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(h 4 ) order for arbitrary quadrilateral meshes when u ∈ H 5 (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(h 4 ) for eigenvalue is derived. Finally, some numerical results are provided to verify the theoretical analysis.

New estimates of mixed finite element method for fourth‐order wave equation

“…For the nonconforming quasi-Wilson element [18,19] and modified quasi-Wilson element [20,21] [33,34]). Therefore, the method and elements employed in our paper have significant advantages.…”