A New Superconvergence Property of Wilson Nonconforming Finite Element

In this paper the Wilson nonconforming nite element method is considered to solve the general two-dimensional second-order elliptic boundary value problems. A new superconvergence property at the vertices and the midpoints of four edges of rectangular meshes is obtained. The Wilson nonconforming nite element has been widely used in computational mechanics and structural engineering because of its good convergence behavior. In many practical cases,it seems better than the bilinear conforming element.However,it is shown in 2],,3],,4]that the convergence rate of Wilson rectangular element in the energy norm is of rst order. As for the arbitrary quadrilat-eral meshes,a rst-order convergence can also be retained provided a slight restriction on meshes is satissed,seee6]. Furthermore,the rst author has given an examplee7] showing that the rst-order convergence is optimal.Recently,Chen and Lii1] strictly proved this rst-order optimality. Meanwhile,computations have observed its superconvergence at the center of elements, thus the question of superconvergence was raised,seee7],,9]. Li justiied in 5] this observation for the simplest model: ?4u = f. Subsequently,the result was extended to the general second-order elliptic problems in 1], see Lemma 2. On the other hand,following 8],it can be easily proved that the bilinear conforming element posseses the superconvergence at the center, as well as at the four vertices and the midpoints of four edges of rectangular meshes. In this paper we prove that besides the center of rectangles as shown in 1], Wilson element has also the superconvergence at these eight points like the bilinear element. We consider the general second-order elliptic boundary value problem