Singularly perturbed equations, in which a small parameter multiplies the highest derivative term, are model equations for many processes such as convection dominated flows. A number of methods in the literature seek to satisfy a criteria known as uniform convergence, in which the error constants are independent of the small parameter as well as the mesh size. One class of such methods involves the use of a refined mesh in the boundary layer(s). We consider the use of multigrid methods for the solution of the difference schemes arising from such discretizations. In the case where the difference scheme uses upwinded differences, the resulting matrix can be highly non-symmetric.