MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1147–1177 S 0025-5718(03)01486-8 Article electronically published on February 3, 2003 FINITE ELEMENT SUPERCONVERGENCE ON SHISHKIN MESH FOR 2-D CONVECTION-DIFFUSION PROBLEMS ZHIMIN ZHANG Abstract. In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N -2 ln 2 N + N -1.5 ln N) in a discrete -weighted energy norm is established under certain regularity assumptions. This conver- gence rate is uniformly valid with respect to the singular perturbation param- eter . Numerical tests indicate that the rate O(N -2 ln 2 N) is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the L 2 -norm. Furthermore, under the same regularity assumption, an -uniform convergence of order N -3/2 ln 5/2 N + N -1 ln 1/2 N in the L ∞ norm is proved for some mesh points in the boundary layer region. 1. Introduction There has been extensive research in numerical solutions of singular perturbation problems because of the practical importance of these problems (for example, the Navier-Stokes equations at high Reynolds number). One of the typical behaviors of singularly perturbed problems is the boundary layer phenomenon: the solution varies rapidly within very thin layer regions near the boundary. Most of the traditional numerical methods fail to catch the rapid change of the solution in boundary layers, and this failure in turn pollutes the numerical approximation on the whole domain. See [18] and [22]. Many methods have been developed to overcome the numerical difficulty caused by boundary layers. The reader is referred to three 1996 books [13, 14, 16] for the significant progress that has been made in this field, and articles [2, 4, 7, 8, 11, 12, 15, 18, 19, 20, 21, 24, 25] for more information. A realistic approach in practice may be starting with a certain up-winding scheme, such as the streamline-diffusion method, followed by an adaptive proce- dure to refine the mesh, eventually resolving the boundary layer, and maybe locat- ing some possible internal layers. Then a question arises naturally: Is there any superconvergence phenomenon when the boundary layer is successfully resolved? The current work intends to answer this question for a specific situation. Received by the editor July 19, 2000 and, in revised form, December 10, 2001. 2000 Mathematics Subject Classification. Primary 65N30, 65N15. Key words and phrases. Convection, diffusion, singularly perturbed, boundary layer, Shishkin mesh, finite element method. This research was partially supported by the National Science Foundation grants DMS- 0074301, DMS-0079743, and INT-0196139. c 2003 American Mathematical Society 1147 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use