Evolutional factorization and superfast relaxation count

In finite-difference solution of multi-dimensional elliptic equations the systems of linear algebraic equations with strongly rarefied matrices of enormous sizes appear. They are solved by iteratonal methods with slow convergence. For rectangular nets, variable coefficients and net steps much more fast method is proposed. In case of finite difference schemes for parabolic equations an efficient method, called evolutional factorization, is built. For elliptic equations relaxation count for evolutionally factorized schemes is proposed. This iterational method has logarithmic convergence. A set of steps, that practically optimizes the method's convergence, and Richardson-like procedure of steps regulation are proposed. The procedure delivers an a posteriori asymptotically precise estimation for the iterational process error. Such estimations were not known before.