On solving a special class of weakly nonlinear finite-difference systems

In this paper, we consider the Newton-iterative method for solving weakly nonlinear finite-difference systems of the form F(u)=Au+G(u)=0, where the jacobian matrix G'(u) satisfies an affine invariant Lipschitz condition. We also consider a modification of the method for which we can improve the likelihood of convergence from initial approximations that may be outside the attraction ball of the Newton-iterative method. We analyse the convergence of this damped method in the framework of the line search strategy. Numerical experiments on a diffusion-convection problem show the effectiveness of the method.