Evolutionary factorization and superfast relaxation count

Abstract

Systems of linear algebraic equations with strongly rarefied matrices of enormous size appear in finite-difference solving of multidimensional elliptic equations. These systems are solved by iteration methods that converge rather slowly. For rectangular grids with variable coefficients and net steps, a considerably faster method is proposed. In the case of difference schemes for parabolic equations, a cost effective method called evolutionary factorization is developed. For elliptic equations, a relaxation count by evolutionally factorized schemes is suggested. This iteration method has the logarithmic convergence speed. A set of steps that practically optimizes the convergence of this algorithm and a procedure to regulate the steps that resembles the Richardson method are proposed. The procedure allows one to obtain an a posteriori asymptotically accurate error estimate of the iteration process. Previously such estimates for iteration processes were not known.