Name: Journée Analyse Appliquée Hauts-de-France
Start: 2024-06-04T09:00:00+02:00
End: 2024-06-04T18:00:00+02:00
Location: Université de Technologie de Compiègne

09:00

Café d'accueil des participants

François Alouges: Natation optimale à bas nombre de Reynolds

La natation à bas nombre de Reynolds est le royaume de prédilection des microorganismes tels que les bactéries ou les spermatozoïdes.
Dans cet exposé, nous détaillerons les méthodes de base qui permettent de contourner le fameux Théorème de la coquille Saint-Jacques
de E. M. Purcell grâce à des boucles dans l’espace des formes du nageur. Puis nous exposerons des résultats récents permettant
d’appréhender des situations où le nombre de contrôles est supérieur à deux et qui permettent de comprendre la forme des brassées
optimales de certains nageurs artificiels.

Claire Chainais-Hillairet: Corrosion of iron in an underground repository: a new thermodynamically consistent model and some theoretical and numerical results

The modelling and the numerical simulation of corrosion take part in the general description of the nuclear waste repository. The derivation of models that are accurate in the long-time regime is a challenge, especially in this context. In this talk, I will start by recalling the Diffusion Poisson Coupled Model introduced by Bataillon et al. en 2010 and I will show how some minor corrections lead to a thermodynamically consistent model. This model consists in a drift-diffusion-Poisson system of equations on a moving domain. I will review the mathematical results we recently obtained for this model and the main issues we are currently considering.

12:00

Déjeuner

André Harnist: Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems

In this talk, we present a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying Lipschitz-continuity and strong-monotonicity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the energy difference that depend locally on the nonlinearity and improve the usual bound. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.

Adina Ciomaga: Homogenization of nonlocal Hamilton Jacobi equations

I will present the framework of periodic homogenisation of nonlocal Hamilton-Jacobi equations, associated with Levy-Itô integro-differential operators. A typical equation is the fractional diffusion coupled with a transport term, where the diffusion is only weakly elliptical. Homogenization is established in two steps: (i) the resolution of a cellular problem - where Lipshitz regularity of the corrector plays a key role and (ii) the convergence of the oscillating solutions towards an averaged profile - where comparison principles are involved. I shall discuss recent results on the regularity of solutions and comparison principles for nonlocal equations, and the difficulties we face when compared with local PDEs. The talk is based on recent results obtained in collaboration with D. Ghilli, E. Topp and O.Ley, E. Topp, T. Minh Le.

16:00

Café