Hussein ALBAZZAL : A posteriori error analysis for some problems related to fuel cell simulations"

Abstract : The main motivation of this thesis is the need for efficient numerical simulations of the gas flows in the serpentine channels of the proton-exchange membrane fuel cells. We consider Poisson and Stokes models in a 2D domain which is composed of several long straight rectangular sections and several bends. In order to speed up the resolution and to reduce the computational costs, we propose 0D models (a fixed parabolic profile for Poisson equation and Poiseuille flow for Stokes equations), and we apply a finite element resolution for the 2D model in the bends. In order to achieve the desired tolerance of the error between the exact solution and the approximated solution coming from the 0D/2D coupled model, we have to overcome a double challenge: how to detect the suitable position of the interface between the 0D and 2D models and how to control the discretization error in the bends. For this purpose, we have developed a posteriori error estimators based on equilibrated flux reconstruction in the subdomains where the finite element method is applied. In the coupled 0D/2D model for Poisson, the estimates give a global guaranteed computable upper bound of the energy norm of the solution. In the coupled 0D/2D model for Stokes, the estimates give a global guaranteed upper bound for the H1-error in velocity and the L2-error in pressure on the whole domain. In the latter case, the estimator involves the inf-sup constant which is in general unknown (in our test cases, it is known to be very small though), thus making the estimator not completely computable in practice. We have also studied the influence of the inf-sup constant on the efficiency of a posteriori error estimates and we have pursued some ideas to construct new guaranteed estimators which are independent of the inf-sup constant. Global lower bounds for the error are also derived for Poisson and Stokes model. The proposed estimators can be split into two parts: a first one indicating the error due to the position of the interface and a second one indicating the error due to the discretization. Using these estimators, an algorithm is proposed to choose the interface position and to make adaptive mesh refinement in order to balance the two sources of the error and to achieve the desired accuracy. The estimators and the adaptive algorithm are validated numerically.

Jury Composition : 

Alexei LOZINSKI, Professeur, Université Bourgogne Franche-Comté, PhD Director

Roberta TITTARELLI, Maîtresse de conférences, SUPMICROTECH-ENSMM, Co-PhD Director

Martin VOHRALIK, Directeur de recherche, Inria Paris, Reporter

Jérôme  POUSIN, Professeur, Université de Lyon, Reporter

Véronique   MARTIN, Maîtresse de conférences, Université de Picardie Jules Verne, Reviewer

Toni SAYAH, Professeur, Université Saint Joseph de Beyrouth, Liban, Reviewer

Localization : Amphithéâtre JJ Gagnepain, TEMIS, 15B Avenue des Montboucons - Besançon