Parameter-Uniform Numerical Methods for a Class of Singularity Perturbed Problems with a Neumann Boundary Condition

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed model ordinary differential equations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. For the same classical finite difference operator on an appropriate piecewise–uniform mesh, it is shown that the numerical solutions converge, uniformly with respect to the perturbation parameter, to the exact solution of any problem from this class.