Local projection stabilisation for higher order discretisations of convection-diffusion problems on Shishkin meshes

Abstract

We consider a singularly perturbed convection-diffusion equation on the unit square where the solution of the problem exhibits exponential boundary layers. In order to stabilise the discretisation, two techniques are combined: Shishkin meshes are used and the local projection method is applied. For arbitrary r≥2, the standard Q r -element is enriched by just six additional functions leading to an element which contains the P r+1. In the local projection norm, the difference between the solution of the stabilised discrete problem and an interpolant of the exact solution is of order \(\mathcal{O}\big((N^{-1}\ln N)^{r+1}\big),\) uniformly in e. Furthermore, it is shown that the method converges uniformly in e of order \(\mathcal{O}\big((N^{-1}\ln N)^{r+1}\big)\) in the global energy norm.