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Evolutionary factorization and superfast relaxation count

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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.

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Correspondence to A. A. Belov.

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Original Russian Text © A.A. Belov, N.N. Kalitkin, 2014, published in Matematicheskoe Modelirovanie, 2014, Vol. 26, No. 9, pp. 47–64.

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Belov, A.A., Kalitkin, N.N. Evolutionary factorization and superfast relaxation count. Math Models Comput Simul 7, 103–116 (2015). https://doi.org/10.1134/S2070048215020039

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