APPROXIMATE NEWTON METHOD FOR COUPLED NONLINEAR SYSTEMS.

We propose an approximate Newton method for solving the coupled nonlinear system G(u, t) equals 0 and N(u, t) equals 0. The method involves applying the basic iteration S of a general solver for the equation G(u, t) equals 0, with t fixed. It is therefore well suited for problems for which such a solver already exists or can be implemented more efficiently than a solver for the coupled system. We derive conditions for S under which the method is locally convergent. The results are applied to continuation methods where N represents a pseudo-arclength condition. We show that under certain conditions the algorithm converges if S is convergent for G. Numerical results are given for a two-level nonlinear multi-grid solver applied to a nonlinear elliptic problem.