A fourth order compact finite difference scheme and a multigrid method are employed to solve the two dimensional convection diffusion equations with boundary layers. The computational domain is first discretized on a nonuniform (stretched) grid to resolve the boundary layers. A grid transformation technique is used to map the nonuniform grid to a uniform one. The fourth order compact scheme is applied to the transformed uniform grid. A multigrid method is used to solve the resulting linear system. We show how the grid stretching affects the computed accuracy of the solutions from the fourth order compact scheme and how the grid stretching influences the convergence rate of the multigrid method. Numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and discretized by the fourth order compact scheme.