The sdfem on Shishkin meshes for linear convection-diffusion problems

Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local $L_\infty$ error estimates that hold true uniformly in the perturbation parameter $\varepsilon$, provided only that $\varepsilon \le N^{-1}$, where ${\cal O}(N^2)$ mesh points are used. Numerical experiments support these theoretical results.