A Two–Stage Iterative Method for Solving Weakly Nonlinear Systems ∗ Emanuele Galligani Dipartimento di Matematica Universit`a di Modena e Reggio Emilia Via Campi 213/b, 41100, Modena, Italy e–mail: galligani@unimo.it Abstract In this paper we consider a two–stage iterative method for solving weakly nonlinear systems generated by the discretization of semilinear elliptic boundary value problems. This method is well suited for imple- mentation on parallel computers. Theorems about the convergence and the monotone convergence of the method are proved. An application of the method for solving real practical problems related to the study of reaction–diffusion processes and of interacting populations is described. Key Words: Newton method, Arithmetic Mean method, monotone con- vergence. 1 The Modified Newton–Arithmetic Mean Method Let F : R n → R n be a continuously differentiable mapping, with an invertible jacobian matrix F ′ (u) for u ∈ R n . This paper will consider the problem of solving large systems of n nonlinear equations F (u)=0 (1) where the matrix F ′ (u) is sparse. In many problems of practical interest the mapping F (u) has the form F (u)= Au + G(u) (2) where A is a nonsingular matrix and G(u) is a continuously differentiable map- ping with G ′ (u) invertible for all u ∈ R n . * This work was communicated at the conference on “High Performance Scientific Comput- ing”, in honor of Professor David John Evans, Bologna, February 5–6, 2001. 1