The green function and A priori estimates of solutions of monotone three-point singularly perturbed finite-difference schemes

We consider a two-point boundary value problem for a singularly perturbed convection-diffusion equation in divergent and nondivergent forms, The numerical solution of the problem is performed with the use of a monotone three-point scheme. On an arbitrary nonuniform grid, we obtain uniform two-sided a priori estimates for the grid solution and its first-order divided difference multiplied by a small parameter in the uniform norm via the corresponding negative norm of the right-hand side. The above-mentioned a priori estimates are derived with the use of the Green function, for which we also establish appropriate estimates in the corresponding anisotropic norms. Our a priori estimates can be used to justify the uniform convergence of various finite-difference schemes under quite general assumptions about the grid, but we do not discuss these problems here.