# Monotonic convergence of Newton's method for boundary value problems [closed]

I’m interested in solving nonlinear elliptic boundary value problems of the type $$-a\Delta u + f(u) = 0,$$ $$u|_\Gamma = u_0$$ by Newton’s method when its convergence is global and monotonic. Could you advice some references concerning this problem, containing proofs of global convergence?

Newton's method takes the form $$-a\Delta u + f(\widetilde u) + f'(\widetilde u)(u - \widetilde u) = 0$$ where $\widetilde u$ is the previous approximation for the solution.