Uniform convergence with respect to a small parameter of a scheme with central difference on refining grids

For a singularly perturbed one-dimensional time-independent divergence equation of diffusion-convection, a scheme is analyzed that approximates the first-order derivative by the central difference. It is proved that this scheme is uniformly convergent with respect to a small parameter in the difference norm L_∞^h on the Bakhvalov and Shishkin grids refined in the boundary layer; the convergence rate is O(N^-2) and O(N^-2ln²N), respectively, where N is the number of grid points. The smoothness condition on the Bakhvalov grid is replaced by a weaker condition.