A Priori Estimates for the Solution of Convection-Diffusion Problems and Interpolation on Shishkin Meshes

Abstract

The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions $n=1$ and $n=2$, sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in $L_{\infty }$ and in the energy norm.