The Research On Convergence Of Nonlinear Multisplitting Algorithms

The system of nonlinear equationsF(x) = 0 (*)has been widely applied in physics, mechanics and engineering. With the development of science and computing technique, many algorithms for solving the system of nonlinear equations have been constructed and analyzed. Especially with the development of parallel computing technique, corresponding numerical and theoretic-researches have made great progress.Multisplitting methods, which have been widely studied in recently years, are a class of numerical computing methods having a very good parallel structure. As a result, they are easily to be used to calculate in parallel. With the development of parallel technique, the study of these methods has been greatly stimulated. As for the boundary-value problems of linear differential equations, multisplitting methods have been extensionally studied. The theories concerned with the convergence as well as the convergent rate analysis have been established systematically. On the other hand, as for the boundary-value problems for nonlinear differential equations, both the algorithmic construction and the theoretical analysis of multisplitting methods are needed. Researches in the fields are relatively few.In this paper, we firstly present a multisplitting additive Schwarz algorithm and a two-level multisplitting additive Schwarz algorithm for solving a system of nonlinear equationsF(x) = Ax - G(x) - b = 0, (**)where A R is a nonsingular matrix, 6 R is a given vector and G is a given nonlinear mapping. This kind of algorithms, combining multisplitting and additive Schwarz algorithm, are easily to be applied to compute in parallel. The convergence and convergent rate is analyzed.Secondly, considering there are many local convergence theories of nonlinear multisplitting methods, we emphatically point out the global convergence and unilateral convergence of solving a kind of relatively special nonlinear system of equations. We also consider the convergence of a nonlinear multisplitting method, in which the solution of the subproblems are approximated by the iterative solution such as m-steps Newton's iterative solution.IIIn the end. numerical examples based on the algorithms presented in this paper are given and the results show these algorithms are effective.