Finite Element Superconvergence on Shishkin Mesh for 2-D Convection-Di usion Problems

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergence 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 convergence rate is uniformly valid with respect to the singular perturbation parameter. 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 1 norm is proved for some mesh points in the boundary layer region.