On the monotone convergence of Jacobi–Newton method for mildly nonlinear systems

Abstract

For mildly nonlinear systems, involving concave or convex diagonal nonlinearities, semi-global monotone convergence of Newton’s method is guaranteed provided that the Jacobian of the system has a nonnegative inverse. However, regardless of this convergence result, the efficiency of Newton’s method becomes poor for stiff nonlinearities. We propose a nonlinear preconditioning procedure inspired by the Jacobi method and resulting in a new system of equations, which can be solved by Newton’s method much more efficiently. The obtained preconditioned method is shown to be globally convergent.