Analysis of a Galerkin finite element method on a Bakhvalov–Shishkin mesh for a linear convection–diffusion problem

We consider a Galerkin finite element method that uses piecewise bilinears on a modified Shishkin mesh for a model singularly perturbed convection–diffusion problem on the unit square. The method is shown to be convergent, uniformly in the perturbation parameter ε, of order N⁻¹in a global energy norm, provided only that ε ≤ N⁻¹, where O(N²)mesh points are used. Thus on the new mesh the method yields more accurate results than on Shishkin’s original piecewise uniform mesh, where it is convergent of order N⁻¹lnN. Numerical experiments support our theoretical results.