## ABSTRACT

This chapter is devoted to various difference approximations of second-order elliptic equations. In Sections 1-3 we present results of more a detailed exploration of the Dirichlet difference problem for Poisson's equation. The approximation technique for the Laplace operator and formulations of difference boundary conditions are described for regions of arbitrary shape. The maximum principle (Section 2) and all of its corollaries are established for grid equations of common structure. These tools are aimed at establishing the uniform convergence with the rate 0(\ h 2) for the difference scheme constructed in Section 1 for the case of an arbitrary domain. In Section 4 we study the properties of the difference Laplace operator and develop the difference operators corresponding to elliptic operators of general form with variable coefficients': In Section 5 higher-accuracy schemes are designed for Poisson's equation in a rectangle.