The paper presents the method of constructing the difference analogs of elliptic operators based on the use of their factorized structure (for second order equations as an example). For the Poisson equations the structure of the difference operators obtained allows us to suggest a new efficient method for solving the difference problems in the domains of standard shape – the method of part-by-part inversion. The number of operations for obtaining a solution by this method coincides, in its order, with the number of operations necessary to realize conventional efficient direct methods: sweeping, fast Fourier transform, cyclic reduction technique.