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

I’m interested in solving nonlinear elliptic boundary value problems of the type $$-a\Delta u + f\left(u\right) = 0, \\ u\big\vert_\Gamma = u_0$$ 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 $$-a\,\Delta u + f\left(\widetilde u\right) + f'\left(\widetilde u\right)\left(u - \widetilde u\right) = 0$$ where $\widetilde u$ is the previous approximation for the solution.

• Cross-posted: scicomp.stackexchange.com/q/20147/713 – Kirill Jul 13 '15 at 6:48
• I've found an article by Schryer "Newton's method for convex nonlinear elliptic boundary value problem" but it concerns strong solutions. Are there articles concerning weak solutions? – jokersobak Oct 17 '15 at 21:59
• @jokersobak You should edit the additional information you have found into the question if it belong there, or write it as an answer if it answers the question. – Tommi Brander May 18 '18 at 12:29