We review and extend results on the local convergence of the classical Newton-Kantorovich method. Then we discuss globally convergent damped and inexact Newton methods and point out advantages of using a minimal error conjugate gradient method for the linear systems arising at each Newton step.
Finally application on a nonlinear elliptic problem is considered. A combination of nested iterations, damped inexact Newton method and two-level grid finite element methods for the solution of the linear boundary value problems encountered at each step are discussed.