A FINER MESH INDEPENDENCE OF NEWTON'S METHOD

Abstract

This study is motivated by optimization considerations and presents a finer thanbefore [1]-[9], [11] affine invariant theory on the asymptotic mesh independence of Newton'smethod for discretized nonlinear operator equations. Using a weaker version of the Newton-Mysovskikh theorem [10], [7] (NMT), for finding solutions of nonlinear operator equationsin a Banach space setting we provide a finer mesh independence of Newton's method withthe following advantages over earlier works [1]-[9], [11]: (a) The usage of mesh independenceis extended under weaker hypotheses; (b) error bounds are finer; (c) the information on thelocation of the solution is more precise; (d) Lipschitz constants are smaller; (e) discretization algorithms are faster and terminate earlier.