**Date & Time:** Thursday, June 23, 2011,
15:15-16:00

**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.