Local projection stabilisation on S-type meshes for convection–diffusion problems with characteristic layers

Abstract

Singularly perturbed convection–diffusion problems with exponential and characteristic layers are considered on the unit square. The discretisation is based on layer-adapted meshes. The standard Galerkin method and the local projection scheme are analysed for bilinear and higher order finite element where enriched spaces were used. For bilinears, first order convergence in the e-weighted energy norm is shown for both the Galerkin and the stabilised scheme. However, supercloseness results of second order hold for the Galerkin method in the e-weighted energy norm and for the local projection scheme in the corresponding norm. For the enriched $${\mathcal{Q}_p}$$-elements, p ≥ 2, which already contain the space $${\mathcal{P}_{p+1}}$$, a convergence order p + 1 in the e-weighted energy norm is proved for both the Galerkin method and the local projection scheme. Furthermore, the local projection methods provides a supercloseness result of order p + 1 in local projection norm.