Nyo-7689 of arbitrarily high order tends to the corresponding deriva tive. Nowhere do we assume the existence of the solution to the differential equation problem on the contrary, we obtain a simple existence proof by using the limiting processi. For the case of elliptic equations convergence is obtained independently of the choice of mesh, but we will find that for the case of the initial value problem for hyperbolic equations, convergence is obtained only if the ratio of the mesh widths in different directions satis fies certain inequalities which ih turn depend on the position of the characteristics relative to the mesh.