The rate of uniform convergence with respect to a small parameter of a difference scheme for an ordinary differential equation

The classical difference scheme on a piecewise-uniform grid that clusters in the boundary layer, approximating the boundary-value problem for an ordinary self-adjoint second-order differential equation with a small parameter in the highest derivative, is investigated. In the grid norm ∥.∥ L ∞ h , the uniform estimate of the rate of convergence with respect to the small parameter that is obtained is O(N -2 ln 2 N), where Nis the number ofgrid nodes. An a priori estimate is obtained using Green's grid function.