Local projection methods on layer-adapted meshes for higher order discretisations of convection-diffusion problems

We consider singularly perturbed convection-diffusion problems in the unit square where the solutions show the typical exponential layers. Layer-adapted meshes (Shishkin and Bakhvalov-Shishkin meshes) and the local projection method are used to stabilise the discretised problem. Using enriched Q(r)-elements on the coarse part of the mesh and standard Q(r)-elements on the remaining parts of the mesh, we show that the difference between the solution of the stabilised discrete problem and a special interpolant of the solution of the continuous problem convergences F-uniformly with order O(N-(r+1/2)) on Bakhvalov-Shishkin meshes and with order O(N-(r+1/2) + N-(r+1) Inr+3/2 N) on Shishkin meshes. Furthermore, an epsilon-uniform convergence in the epsilon-weighted H-1-norm with order O((N-1 In N)(-r)) on Shishkin meshes and with order O(N-r) on Bakhvalov-Shishkin meshes will be proved. Numerical results which support the theory will be presented. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.