Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers

Abstract

On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has two exponential boundary layers. We apply the streamline-diffusion finite element method with piecewise bilinear trial functions on a Shishkin mesh of O(N²) points and show that the error in the discrete space between the computed solution and the interpolant of the true solution is, uniformly in the diffusion parameter ɛ, of order ɛ^1/2N^–1 lnN + N^–3/2 in the usual streamline-diffusion norm. This includes an L²-norm error estimate of order O(N^–3/2) in the convection–dominated case ɛ ⩽ N^–1 ln^–2N. As a corollary we prove that the method is convergent of order N^–1/2 ln^3/2N (again uniformly in ɛ) in the local L^∞ norm on the fine part of the mesh (i.e., inside the boundary layers). This local L^∞ estimate within the layers can be improved to order ɛ^1/2N^–1/2 ln^3/2N+N^–1 ln^1/2N, uniformly in ɛ, away from the corner layer.