J. Numer. Math., Vol. 12, No. 2, pp. 77–96 (2004) c VSP 2004 Interior superconvergence of nite element solutions for Stokes problems by local L 2 projections H. Chen Received December 17, 2002 Communicated by Z. Chen Received in revised form October 20, 2003 Abstract — This paper derives some interior superconvergenceestimates for nite element solutions of the Stokes problem by a local L 2 projection method. The results depend only on local properties of the Stokes problem and the nite element approximations. Keywords: interior error estimate, nite element method, superconvergence, L 2 projection method, Stokes problem 1. INTRODUCTION In this paper, we are concerned with interior estimates of superconvergence for the nite element approximation of the Stokes equation by a local L 2 projection method. The L 2 projection method is a new superconvergence technique developed recently in Wang [23], Chen and Wang [6] and Heimsund, Tai and Wang [11] for second order elliptic problems. A favorable feature of the L 2 projection method is that it provides superconvergence for general nite element partitions, while the traditional superconvergence results require some stringent conditions on the nite element partition, such as certain uniformity, or symmetry about a point or tensor product of one-dimensional meshes, etc. We refer to K† r´ ‡† zek and Neittaanm¨ aki [12], Zhu and Lin [27], or Wahlbin [22] and the literature cited there for a review. For the Stokes problem, the existing superconvergence results of nite element solutions are all based on uniform or rectangular partitions and are only for special choices of nite element spaces (see, e.g., Lin, Li and Zhou [16], Lin and Pan [17] and Pan [20]). Global superconvergence of nite element solutions of Stokes problem for general meshes by L 2 projection method has been obtained in Wang and Ye [24]. Although the results in [24] no longer assume any uniformity, they still rely strongly on the smoothness of the true solution and a high a priori regularity of the Stokes problem globally over the whole domain. In general, the required a priori regularity holds true for problems with sufciently smooth data and domain. Consequently, the superconvergence results developed in [24] has a theoretical limitation in practical applications. Department of Mathematics, University of Wyoming, Laramie, WY 82070