For nonlinear infinite dimensional equations, which to begin with: Linearization or discretization?

Abstract

To tackle a nonlinear equation in a functional space, two numerical processes are involved: discretization and linearization. In this paper we study the differences between applying them in one or in the other order. Linearize first and discretize the linear problem will be in the sequel called option~(A). Discretize first and linearize the discrete problem will be called option~(B). As a linearization scheme, we consider the Newton method. It will be shown that, under certain assumptions on the discretization method, option~(A) converges to the exact solution, contrarily to option~(B) which converges to a finite dimensional solution. These assumptions are not satisfied by the classical Galerkin, Petrov-Galerkin and collocation methods, but they are fulfilled by the Kantorovich projection method. The problem to be solved is a nonlinear Fredholm equation of the second kind involving a compact operator. Numerical evidence is provided with a nonlinear integral equation.