I will speak about the project "Parallel algorithms for non linear iterative methods" realized during CEMRACS 2016. The simulation of complex non linear structures via the finite element method is generally based on quasi-static incremental loading leading to successive non linear problems solved by Newton-like methods which themselves entail the solution of multiple ill-conditioned large linear systems. For complex structures, the total number of steps may be very large, and, since the procedure is recursive, the time to solution is very long. Reducing the time for such simulations is of great interest for numerical engineering in industry. The main goal for us was to derive algorithms from the Parareal [J.L. Lions, Y. Maday and G. Turinici, 332(7), pp. 661-668, 2001] parallel in time method in order to perform these steps in parallel.
This is joint work with Yvon Maday, François-Xavier Roux and Mustafa Gaja

[1] E. Brivadis, A. Buffa, B. Wohlmuth and L. Wunderlich, Isogeometric mortar methods, Computer Methods in Applied Mechanics and Engineering, 284 (2015), pp. 292-319.
[2] L. De Lorenzis, P. Wriggers and G. Zavarise, A mortar formulation for 3d large deformation contact using nurbs based isogeometric analysis and the augmented Lagrangian method, Springer-Verlag, 49 (2012), pp. 1-20.
[3] A. E. Maliki, M. Fortin, J. Deteix and A. Fortin, Preconditioned iteration for saddle-point systems with bound constraints arising in contact problems, Computer Methods in Applied Mechanics and Engineering, 254 (2013), pp. 114-125.

Dans cet exposé, on va présenter premièrement le développement et l’analyse d’une méthode numérique efficace, précise et plus simple et à mettre en œuvre que cette méthode existe pour la résolution d’un système d’équations différentielles ordinaires linéaires et non linéaires. Deuxièmement, on propose un nouveau modèle d’ordre fractionnaire pour la simulation de la dynamique de l’épidémie de l’obésité. En utilisant la méthode d’ondelettes de Legendre combinée à la technique de découplage et de quasi-linéarisation, on démontre la validité et l’applicabilité de notre modèle. On présente également quelques exemples illustratifs fractionnaires pour démontrer l’applicabilité et l’efficacité du procédé. La dérivée fractionnaire est décrite au sens de Caputo .
On termine par la construction d’un nouveau modèle mathématique classique des mécanismes moléculaires des complications métaboliques liées à l’obésité dont en particulier les problèmes cardiovasculaire, le diabète de type 2 et le cancer pour passer à l’étape de validation en utilisant des données expérimentales.

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modelled by a standard normal compliance condition without unilateral constraint. For each problem, we derive a variational formulation and an existence result of the weak solution of the regularized problem is obtained. Next, we prove the convergence of the weak solution of the regularized problem to the weak solution of the initial nonregularized problem. Then, we introduce a fully discrete approximation of the variational problem based on a finite element method and on a second order time integration scheme. The solution of the resulting nonsmooth and nonconvex frictional contact problems is presented, based on approximation by a sequence of nonsmooth convex programming problems. Finally, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the convergence result.

A new class of Schwarz methods was recently presented in the literature for the solution of solvation models, where the electrostatic energy contribution to the solvation energy can be computed by solving a system of elliptic partial differential equations [1,2]. Numerical simulations have shown an unusual convergence behaviour of Schwarz methods for the solution of this problem, where each atom corresponds to a subdomain : the convergence of the Schwarz methods is independent of the number of atoms [1], even though there is no coarse grid correction. Despite the successful implementation of Schwarz methods for this solvation model, a rigorous analysis for this unusual convergence behaviour is required, since no theoretical results are given in the corresponding literature.
In this talk, we analyze the behavior of the Schwarz method for the solution of a chain of atoms and show that its convergence does not depend on the number of atoms (subdomains). We use two different techniques to prove this result. The first technique is based on a Fourier expansion of the error and the analysis of transfer matrices constructed for an approximate model. The second one consists in an application of the maximum-principle and allows us to analyze very general geometries.
[1] Cancès et al., Domain decomposition for implicit solvation models, J. of Chem. P. (2013).
[2] Lipparini et al., Fast Domain Decomposition Algorithm for Continuum Solvation Models, J. Chem. Theory Comput. (2013).

Let be a bounded domain of class . In , we consider a self-adjoint second order elliptic differential operator with the Dirichlet boundary condition. The coefficients of are periodic and depend on  ; so, they oscillate rapidly as . We obtain approximations for the resolvent and for the semigroup , , both in the and norms. The results of such type are called operator error estimates in homogenization theory.

The talk is based on a joint work with T. A. Suslina.

On propose un modèle microscopique de mouvement de foule basé sur la théorie des jeux. On suppose que chaque individu est influencé par certains de ses voisins, par exemple ceux qui sont dans son cône de vision, et on représente ces relations d’influence par un graphe. Les vitesses actuelles des individus résultent d’un équilibre de Nash instantané où chaque individu essaye de faire son mieux en minimisant une certaine fonction coût personnelle, et en tenant compte du comportement des voisins qui l’influencent (contraintes de non-chevauchement). On étudie théoriquement le modèle pour différents types de graphes. On propose un schéma numérique pour résoudre le problème dans le cas où le graphe d’influence est acyclique. Les résultats numériques sont confrontés à des expériences réelles d’évacuation de piétons. Notamment on prouve la capacité du modèle à reproduire des effets de mouvement de foule observés en réalité comme l’alternance de courts et longs laps de temps entre deux sorties, la loi de puissance pour les grands laps de temps, l’effet Faster is Slower et l’effet bénéfique de la présence d’un obstacle sur le flux sortant.

We introduce a new multi-scale numerical method, handling arbitrary approximation orders, and applicable to general meshes. This method extends the ideas of the multi-scale FE method (MsFEM) à la Crouzeix-Raviart introduced in [1]. MsHHO hinges on the formalism of Hybrid High-Order (HHO) methods, introduced in [2], to operate the order increase and the generalization to elements with arbitrary shape. We prove for the method of order an energy-norm error estimate of the form , where is the micro-scale and the characteristic length of the macro-mesh. One advantage of using the HHO formalism is to allow the addition of supplementary cell-based degrees of freedom without having to compute more oscillatory basis functions. This aspect can be attractive for example in the treatment of perforated domains. More details are available in [3].

This is a joint work with Matteo Cicuttin and Alexandre Ern (CERMICS, École des Ponts ParisTech)

[1] C. Le Bris, F. Legoll and A. Lozinski. MsFEM à la Crouzeix-Raviart for highly oscillatory elliptic problems. Chinese Annals of Mathematics, Series B, 34(1):113–138, 2013.
[2] D. A. Di Pietro, A. Ern and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math., 14(4):461–472, 2014.
[3] M. Cicuttin, A. Ern and S. Lemaire. A Multi-scale Hybrid High-Order method. Available at https://hal.archives-ouvertes.fr/hal-01467434, 2017.