CONVERGENCE AND SUPERCONVERGENCE OF A NONCONFORMING FINITE ELEMENT ON ANISOTROPIC MESHES

The main aim of this paper is to study the error estimates of a nonconforming flnite element for general second order problems, in particular, the superconvergence properties under anisotropic meshes. Some extrapolation results on rectangular meshes are also discussed. Finally, numerical results are presented, which coincides with our theoretical analysis perfectly. It is well-known that regular assumption or quasi-uniform assumption (10, 13) of flnite element meshes is a basic condition in the analysis of flnite element approxima- tion both for conventional conforming and nonconforming elements. However, with the advances of the flnite element methods and its applications to other flelds and more complex problems, the above regular or quasi-uniform assumption becomes quite a restriction in practice for some problems in the flnite element methods. For example, the solution may have anisotropic behavior in part of the domain, that is to say, the solution varies signiflcantly only in certain directions. Such prob- lems are frequently encountered in perturbed convection-difiusion-reaction equa- tions where boundary or interior layers appear. In such cases, it is more efiective to use anisotropic meshes with a small mesh size in the direction of the rapid variation of the solution and a larger mesh size in the perpendicular direction. Consider a bounded convex domain › ‰ R 2 . Let Jh be a family of meshes of ›. Denote the diameter of an element K and the diameter of the inscribed circle of K by hK and ‰K, respectively. h = max K2Jh hK. It is assumed in the classical flnite element theory