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

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 Pr+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 O((N-1 ln N)(r+1)), uniformly in epsilon. Furthermore, it is shown that the method converges uniformly in e of order O((N-1 ln N)(r+1)) in the global energy norm.