Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters

We study a model linear convection–diffusion–reaction problem where both the diffusion term and the convection term are multiplied by small parameters εd and εc, respectively. Depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain. Sharp bounds for the derivatives of the solution are derived using a barrierfunction technique. These bounds are applied in the analysis of a simple upwind-difference scheme on Shishkin meshes. This method is established to be almost first-order convergent, independently of the parameters εd and εc.  2003 Elsevier Inc. All rights reserved.