Superconvergent Finite Element Methods on a Shishkin Mesh for Convection-Diffusion Problems

. In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergent rate O(N \Gamma2 ln 2 N + N \Gamma3=2 ) in a discrete ffl-weighted energy norm is established under certain regularity assumption. This convergent rate is uniformly valid with respect to the singular perturbation parameter ffl. Numerical tests indicate that the rate O(N \Gamma2 ln 2 N) is sharp for the boundary layer terms. As a byproduct, an ffl-uniform convergence of the same order is obtained for the L 2 -norm. Furthermore, under the same regularity assumption, an ffl-uniform convergence of order N \Gamma3=2 ln 5=2 N + N \Gamma1 ln 1=2 N in the L 1 norm is proved for some mesh points in the boundary layer region. Key Words. convection, diffusion, singularly perturbed, boundary layer, Shishkin mesh, finite element method AMS Subject Classification. 65N30, 65N15 1. Introduction. There has been ex..