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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 ln2 N + eN-1.5lnN) in a discrete e-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter e. Numerical tests indicate that the rate O(N-2ln2 N) is sharp for the boundary layer terms. As a by-product, an e-uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an e-uniform convergence of order N-3/2ln5/2N + eN-1ln1/2N in the L∞ norm is proved for some mesh points in the boundary layer region.