Superconvergence Analysis and Error Expansion for the Wilson Nonconforming Finite Element Hongsen Chen * Bo Li † Abstract In this paper the Wilson nonconforming finite element is considered for solv- ing a class of two-dimensional second-order elliptic boundary value problems. Superconvergence estimates and error expansions are obtained for both uniform and non-uniform rectangular meshes. A new lower bound of the error shows that the usual error estimates are optimal. Finally a discussion on the error behaviour in negative norms shows that there is generally no improvement in the order by going to weaker norms. Keywords: Wilson finite element, rectangular mesh, optimal error estimate, negative norm, superconvergence, error expansion, extrapolation Subject classification: AMS(MOS): 65N30 1 Introduction The Wilson finite element, known as Wilson’s brick in three-dimensional finite element applications, is widely used in computational mechanics and structural engineering, see, e.g., [5] and [17]. The corresponding finite element space consists of piececwise * This work was supported by the National Natural Science Foundation of P. R. China and Deutsche Forschungsgemeinschaft, SFB 359, Germany. † This work is part of the Transitions and Defects in Ordered Materials Project and was supported in part by the NSF through grant DMS 911-1572, by the AFOSR through grant AFOSR-91-0301, by the ARO through grants DAAL03-89-G-0081 and DAAL03-92-G-0003, and by a grant from the Minnesota Supercomputer Institute. 1