Approximation of derivatives in a convection–diffusion two-point boundary value problem Original Research Article

We consider a convection–diffusion two-point boundary value problem in conservative form. To solve it numerically an upwind conservative finite difference scheme is applied. On an arbitrary mesh we prove bounds, which are weighted by the small diffusion coefficient, on the errors in approximating the derivative of the true solution by divided differences of the computed solution. On a slightly less general mesh we prove unweighted bounds on these errors where the mesh is coarse. These bounds are then made more explicit for the particular cases of Shishkin and Bakhvalov meshes. Numerical results are presented that demonstrate the sharpness of our results on these eponymous meshes.