The issue addressed in this paper is resolving the boundary layer in a model convection-diffusion ordinary differential equation boundary value problem involving a small parameter &egr;. The finite difference discretization procedure uses a nonuniform grid designed to achieve convergence uniform in &egr;. The paper relies heavily on extensive prior literature, cited therein. The theoretical formulation presented here unifies and extends previous work. The grid is equidistributed, using a monitor function based on a priori or a posteriori information about the solution, and Linß shows that the theory applies. Numerical examples are provided that exhibit uniform convergence as &egr; decreases, using a discrete adaptive method, but the adaptation is not designed to control the error per se.