Convergence of a modified Samarskij's monotonic scheme on a smoothly condensing grid (Q1571211)

A modified monotonic difference scheme on a nonuniform grid is used for approximating a singular perturbed second-order ordinary differential equation that degenerates into a first-order equation when the small parameter tends to zero. A certain optimal method for condensing the grid in the boundary layer and in the neighborhood of this layer is proposed. It is proved that this scheme converges uniformly with respect to the small parameter. The convergence rate is \(O(N^{-2})\), where \(N\) is the total number of grid points.