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 epsilon, of order N-1 in a global energy norm, provided only that epsilon less than or equal to N-1, where O(N-2) 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-1 In N. Numerical experiments support our theoretical results.