Finite element superconvergence on Shishkin mesh for 2-D 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 superconvergence rate O(N−2 ln N + N−1.5 lnN) in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests indicate that the rate O(N−2 ln N) is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an -uniform convergence of order N−3/2 ln N + N−1 lnN in the L∞ norm is proved for some mesh points in the boundary layer region.