NEWTON ADDITIVE AND MULTIPLICATIVE SCHWARZ ITERATIVE METHODS∗

Abstract. Convergence properties are presented for Newton additive and multiplicative Schwarz iterative methods for the solution of nonlinear systems in several variables. These methods consist of approximate solutions of the linear Newton step using either additive or multiplicative Schwarz iterations, where overlap between subdomains can be used. Restricted versions of these methods are also considered. Numerical experiments on parallel computers are presented, indicating the effectiveness of these methods. Key words. Nonlinear systems. Newton’s method. Additive Schwarz. Multiplicative Schwarz. Iterative methods. Subspace correction. Parallel computing. AMS subject classifications. 65H10, 65N55, 65Y05 1. Introduction and Preliminaries. We are interested in the parallel solution of the system of nonlinear equations F (x) = 0,(1.1) where F is a map F: Rn → Rn, and it is assumed that a solution x ⋆ of (1.1) exists. A well-known method for solving the nonlinear system (1.1) is the classical Newto