The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C1,1 is analyzed. It is shown that the maximum error for a nine-point approximation is of the order of O(h2(|ln h|+1)) as a five-point approximation. This order can be improved up to O(h2) when the nine-point approximation in the grids which are a distance h from the boundary is replaced by a five-point approximation (“five and nine”-point scheme). It is also proved that the class of boundary functions C1,1 used to obtain the error estimations essentially cannot be enlarged. We provide numerical experiments to support the analysis made. These results point at the importance of taking the smoothness of the boundary functions into account when choosing the numerical algorithms in applied problems.