Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems

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 $O(N^{-2}\ln ^2 N + \epsilon N^{-1.5}\ln N)$ in a discrete $\epsilon$-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter $\epsilon$. Numerical tests indicate that the rate $O(N^{-2}\ln ^2 N)$ is sharp for the boundary layer terms. As a by-product, an $\epsilon$-uniform convergence of the same order is obtained for the $L^2$-norm. Furthermore, under the same regularity assumption, an $\epsilon$-uniform convergence of order $N^{-3/2}\ln ^{5/2} N + \epsilon N^{-1}\ln ^{1/2} N$ in the $L^\infty$ norm is proved for some mesh points in the boundary layer region.