We present the analysis of an h version Finite Element Method for the approximation of the solution to convection diffusion problems. The method uses piece-wise polynomials of degree p >= 1, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We consider a model convection diffusion problem, posed on the unit square and establish robust, optimal convergence rates in the energy and in the maximum norm. We also present the results of some numerical computations that illustrate our theoretical findings and compare the proposed method with others found in the literature. (C) 2016 Elsevier Ltd. All rights reserved.