Section: New Results

Mathematical analysis of PDEs

Participants : Muriel Boulakia, Jean-Jerome Casanova, Céline Grandmont.

In [23], we consider a reaction-diffusion equation where the reaction term is given by a cubic function and we are interested in the numerical reconstruction of the time-independent part of the source term from measurements of the solution. For this identification problem, we present an iterative algorithm based on Carleman estimates which consists of minimizing at each iteration strongly convex cost functionals. Despite the nonlinear nature of the problem, we prove that our algorithm globally converges and the convergence speed evaluated in weighted norm is linear. In the last part of the paper, we illustrate the effectiveness of our algorithm with several numerical reconstructions in dimension one or two.

In [25] a coupled system of pdes modelling the interaction between a two–dimensional incompressible viscous fluid and a one–dimensional elastic beam located on the upper part of the fluid domain boundary is considered. A good functional framework to define weak solutions in case of contact between the elastic beam and the bottom of the fluid cavity is designed.It is then prove that such solutions exist globally in time regardless a possible contact by approximating the beam equation by a damped beam and letting this additional viscosity vanishes.