by Stynes, Martin, Tobiska, Lutz.

**Series:** 1998-17, Preprints

- MSC:
- 65N15 Error bounds
- 65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

**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

it is convergent, uniformly in the diffusion parameter $\varepsilon$,

of order $\varepsilon^{1/2} N^{-1}\ln^{3/2} N + N^{-3/2}$ in the usual

streamline-diffusion norm. As a corollary we prove that the method is

convergent of order $\varepsilon^{1/2} N^{-1/2}\ln^{2} N + N^{-1/2}\ln^{3/2}N$

(again uniformly in $\varepsilon$) in the local $L^\infty$ norm on the fine

part of the mesh (i.e., inside the boundary layers). This

local $L^\infty$ estimate within the layers can be improved to order

$\varepsilon^{1/2} N^{-1/2}\ln^{2} N + N^{-1}\ln^{1/2}N$, uniformly in

$\varepsilon$, away from the corner layer.

**Keywords:**

Streamline diffusion, finite element method, singular perturbation,convection-diffusion, Shishkin mesh