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 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 uniformly in ε. Furthermore, it is shown that the method converges uniformly in ε of order in the global energy norm.