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

Abstract

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 ε-uniformly with order O ( N − ( r + 1 / 2 ) ) on Bakhvalov–Shishkin meshes and with order O ( N − ( r + 1 / 2 ) + N − ( r + 1 ) ln r + 3 / 2 N ) on Shishkin meshes. Furthermore, an ε-uniform convergence in the ε-weighted H 1 -norm with order O ( ( N − 1 ln 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.