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

Abstract

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 ∈, of order N -1 in a global energy norm, provided only that ∈ ≤ 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.