On the sharpness of error bounds in connection with finite difference schemes on uniform grids for boundary value problems of ordinary differential equations - Büttgenbach, B.; Esser, H.; Nessel, R. J. (Numerical functional analysis and optimization, 1991)
For linear two-point boundary value problems of ordinary differential equations, some convergence properties of approximate solutions Y h obtained by standard finite difference schemes on uniform grids are discussed. By means of discrete Green's functions a representation of the error Y h -Y in functional dependence on the exact solution Y is employed to prove the sharpness (with regard to the order) of well-known error estimates in terms of moduli of smoothness of derivatives of Y.