New numerical process solving nonlinear infinite-dimensional equations

Abstract

Solving a nonlinear equation in a functional space requires two processes: discretization and linearization. In recent paper Grammant et al. (J Integral Equ Appl 26:413-436, 2014), the authors study the difference between applying them in one and in the other order. Linearizing first the nonlinear problem and discretizing the linear problem will be called option (B). Discretizing first the nonlinear problem and linearizing the discrete nonlinear problem will be called option (C). In this paper, we propose a new numerical process equivalent to the option (B): we linearize first the original nonlinear problem with an alternative linearization scheme than that used in the option (B), then we discretize the resulting iterative linear equations using a projection method to implement the corresponding finite-dimensional problem. The aims of this new process are intended to get weaker the theoretical assumptions and to give a powerful numerical performance. We give sufficient conditions to deal with the convergence results. Finally, as a numerical application, we solve a system of Fredholm equations of the second kind. The accuracy and efficiency of this process are illustrated in some numerical examples.

New linearization method for nonlinear problems in Hilbert space

“…Kantorovich's genius appears when he gets the correspondence between 1 A (x) in the real case and (A (x)) −1 in the case of a Banach space, where the first derivative is in the classical sense and the second is a Fréchet derivative (see [1,4,6,9]). For the equation…”

“…which is very hard to do and usually ξ k+1 is numerically approximated in each iteration (see [1,6,9]).…”