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 ( N 2 ) 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 1/2 N –1 ln N + N – 3/2 in the usual streamline-diffusion norm. This includes an L 2 -norm error estimate of order O ( N – 3/2 ) in the convection–dominated case ɛ ⩽ N – 1 ln –2 N . As a corollary we prove that the method is convergent of order N –1/2 ln 3/2 N (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 1/2 N –1/2 ln 3/2 N + N –1 ln 1/2 N , uniformly in ɛ , away from the corner layer.
Published Version
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