A geometric multigrid method for solving the convection diffusion equation with boundary layers to sixth order accuracy is presented. A nine point finite difference discretization scheme is used to obtain fourth order accurate solutions on a coarse and a fine grid. Richardson extrapolation is used to increase the order of accuracy to sixth order on the coarse. An iterative smoothing technique is then used to obtain a sixth order solution on the fine grid. The discretization we used allows the grid to be a graded mesh. This is the first time the post extrapolation smoothing technique has been applied to a graded mesh. Numerical results are presented to demonstrate the use of a graded mesh can significantly decrease the maximum error compared to a regular mesh.