Iterative methods for the solution of some nonlinear elliptic difference systems, approximating the first boundary value problem are considered. If h > 0 is the network step in the space of variables x = (x1, x2,…, xp) and 2m is the order of the original boundary value problem, then the iterative methods proposed give solution of accuracy ε with the expenditure of O(|In ɛ| h−(p+m−½)) and O(|In ɛ| |In h| h−p) arithmetic operations in the case of a general region and a rectangular parallelepiped respectively. In the case p = 2 the estimate O(|In ɛ| h−[2+ (m/2)]) is obtained if the region is made up of rectangles with sides parallel to the co-ordinate axes.