Date & Time: Thursday, June 23, 2011,
Venue:Room No. 215, Department of Mathematics
Title:For nonlinear infinite dimensional equations, which to begin with: linearization or discretization?
Speaker:Prof. Mario AHUES La MUSE, University of Lyon, France
Abstract: We consider a complex Banach space X, a nonempty open set O of X, and a (nonlinear) Fr ́chet differentiable operator F : O ⊂ X → X. The problem is to find e φ ∈ O such that F (φ) = 0. Projection method is a well known technique to compute approximate solutions to such an equation. One notices that methods proposed in literature start with a discretization procedure which leads to a nonlinear approximate equation in a finite dimensional linear space, and next, a numerical scheme for nonlinear finite dimensional equations, such as Newton-Kantorovich method is applied. This scheme involves at each step the resolution of an n×n updated linear system. If the nonlinear operator F is sufficiently smooth, we suggest to proceed in the opposite sense: First, linearize the original equation in the infinite dimensional context (again using the Newton-Kantorovich method) and then, solve numerically the underlying sequence of linear equations using the projection method. The main theoretical result is that, for n large enough but fixed, if the Newton method is applied first and the projection process is used at each step of the Newton method, then the sequence of iterates converges to φ, whilst in the opposite case, the sequence of iterates converges to an n-order approximation of φ. We illustrate this results with a quadratic Fredholm equation of the second kind and with a finite element spectral approximation of a bidimensional differential operator.