R-adaptation par l'estimateur d'erreur hiérarchique

International audienceThe aim of this work is to devise a method to determine the optimal position of the nodes in a finite element discretization for a boundary value problem. The node displacement procedure (also called R-adaptation) is a crucial step in a global mesh adaptation procedure. In the present approch, we determine the nodal position by minimizing the approximation error. This error is evaluated using a hierarchical estimator. A numerical test is presented.L'objectif de ce travail est de déterminer la meilleure position des noeuds d'un maillage, utilisé lors de la discrétisation d'un problème aux limites par la méthode des éléments finis. La procédure de déplacement des noeuds (appelé aussi R-adaptation) est une étape importante dans la stratégie globale d'adaptation de maillage. La position optimale des noeuds est déterminée en minimisant l'erreur d'approximation. Pour évaluer cette erreur nous utilisons l'estimateur d'erreur hiérarchique. Un test numérique est présenté

Mortar finite element discretization of a model coupling Darcy and Stokes equations

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.EU Marie CurieMinisterio de Educación y Cienci

Analysis of an anaerobic digestion model in landfill with mortality term

International audienceWe study a mathematical model of anaerobic digestion with biomass recirculation, dedicated to landfill problems, and analyze its asymptotic behavior. We show that the global attractor is composed of an infinity of non-hyperbolic equilibria. For non-monotonic growth functions, this set is non connected, which impacts the performances of the bioprocess

ETUDE EXPERIMENTALE ET MODELISATION NUMÉRIQUE DES TRANSFERTS HYDRIQUES ET DE SOLUTE DANS UN MILIEU POREUX NON SATURE-SATURE

In this work, a mathematical modeling tool was developped to simulate in a simple and fast way, and with an acceptable precision, the water and solute transfers in unsaturated-saturated porous medium. The resolution of this problem in unstationary mode, is based on the use of an mathematical formulation which consists in considering as a single continuum the saturated and the unsaturated zones. The modeling tool is based on two numerical methods: the alternate directions implicit method (ADI) and the finite elements method (finite elements code Freefem++). This code was adapted to the equations used in this study, namely: Richards’ equation to study the water flow in the unsaturated zone, the diffusivity equation expressed in a linear and nonlinear form for the groundwater flow and the transport equation of advection-dispersion type to study the solute transfer. The advantage of the Freefem++ code is that it generates selfadapting grids. This is interesting to analyse the transfers at singular zones, in particular at the capillary fringe. Experiments were carried out on a laboratory physical model, consisting in a sand filled tank of 160 cm length and 50 cm in height. This experimental model enabled to follow the spacetime evolution of the water-table and the propagation of a tracer (Fluorescein) from the infiltration strip to the discharge system of the model. The modeling tool was validated using experimental data measured on the small-scale physical model and some results taken from the literature. It allowed to predict with an excellent precision the water-table position and the propagation of the non-reactive solute in both unsaturated and saturated compartments. A comparative study between the two numerical methods was carried out and showed that even if the finite differences method provides satisfactory results, the use of the finite elements Freefem++ code presents a light advantage and provides slightly more accurate result