Uniform convergence analysis of an upwind finite difference approximations of an homogenous singularly perturbed boundary value problem using grid equidistribution

We derive varepsilon-uniform error estimates for two first-order upwind discretizations of a model inhomogeneous, second-order, singularly perturbed boundary value problem on a non-uniform grid. Here, varepsilon is the small parameter multiplying the highest derivative term. The grid is suggested by the equidistribution of a positive monitor function which is a linear combination of a constant floor and a power of the second derivative of the solution. Our analysis shows how the floor should be chosen to ensure varepsilon-uniform convergence and indicates the convergence behaviour for such grids.