The convergence rate of economical iterative methods for stationary problems of mathematical physics

Abstract

Nonstationary equations or related finite-difference schemes are used for solving stationary problems of mathematical physics. By [1, p. 550; 2, p. 320], a solution of a stationary problem with positive operators can be treated as a limit (as t ~ oc) solution of the corresponding nonstationary problem. There is extensive literature dealing with this problem [1-9]. Economical iterative methods are especially interesting. They include the classical alternating direction method [1-3], various decomposition methods [7, 8], and factorized methods [1, 2], for which the convergence rate have been analyzed in detail and the possibility of increasing the convergence rate by an appropriate choice of the iteration parameters has been indicated [1, 2, 4, 6-10]. Efficient parallel algorithms have been suggested for economical iterative methods. However, the above-mentioned classical methods have a number of disadvantages; namely, the number of decomposition components is restricted, these components must commute, and the convergence rate is not very high. A many-component alternating direction method free of these disadvantages was suggested in [11-13]. The papers [14, 15] deal with the investigation of iterative methods for stationary problems on the basis of lnany-component finite-difference schemes. In the present paper, we analyze the convergence rate of these methods and issues related to the optimal choice of the iteration parameter.