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 Qr-elements on the coarse part of the mesh and standard Qr-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 ε-uniformly with order O(N−(r+1/2)) on Bakhvalov–Shishkin meshes and with order O(N−(r+1/2)+N−(r+1)lnr+3/2N) on Shishkin meshes. Furthermore, an ε-uniform convergence in the ε-weighted H1-norm with order O((N−1lnN)−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.