The purpose of this article is to acquaint the reader with the general
concepts and capabilities of the Difference Potentials Method (DPM).
DPM is used for the numerical solution of boundary-value and some othe
r problems in difference and differential formulations. Difference pot
entials and DPM play the same role in the theory of solutions of Linea
r systems of difference equations on multi-dimensional non-regular mes
hes as the classical Cauchy integral and the method of singular integr
al equations do in the theory of analytical functions (solutions Cauch
y-Riemann system).The application of DPM to the solution of problems i
n difference formulation forms the first aspect of the method. The sec
ond aspect of the DPM implementation is the discretization and numeric
al solution of the Calderon-Seeley boundary pseudo-differential equati
ons. The latter are equivalent to elliptical differential equations wi
th variable coefficients in the domain; they are written making no use
of fundamental solutions and integrals. Because of this fact ordinary
methods for discretization of integral equations cannot be applied in
this case. Calderon-Seeley equations have probably not been used for
computations before the theory of DPM appeared. This second aspect for
the implementation of DPM is that it does not require difference appr
oximation on the boundary conditions of the original problem. The latt
er circumstance is just the main advantage of the second aspect in com
parison with the first one. To begin with, we put forward and justify
the main constructions and applications of DPM for problems connected
with the Laplace equation. Further, we also outline the general theory
and applications: both those already realized and anticipated.