Estimation d’erreur a posteriori et méthodes de stabilisation éléments finis pour les problèmes de convection diffusion réaction

Abstract

The main objective of this thesis is to provide an a posteriori error estimates for stabilized nite element approximations of convection dominated transport problems. In fact we deal with treamline-upwind/Petrove-Galerkin (SUPG), subgrid viscosity (SGV) and algebraic orthogonal subscale (AOS) stabilization. To achieve this goal, we derive symptotically robust and fully robust a posteriori error estimates. Asymptotically robust means that the error estimates are robust with respect to the Péclet number, while fully robust means that the error estimates are uniform with respect to the vanishing diffusion coefficient. The a posteriori error estimates give upper and lower bounds for the error in the energy norm between the exact solution and approximate solution. These error estimators can be used to deduce adaptive strategies. Contributions are presented on three axes. First, we want to obtain a robust hierarchical a posteriori error estimates for(SUPG) stabilization of convection diffusion equation. The estimator is derived without using the saturation assumption or a comparison with residuals. The second axis is dedicated to a posteriori error estimates for (SGV) and (AOS) stabilization of convection diffusion problems. Two a posteriori error estimators are derived. The first one is asymptotically robust and the second one is fully robust. Numerical results in the test cases with boundary or interior layers show that the asymptotically robust estimator can be used to construct adaptive mesh. Finally, we present also the construction of two a posteriori error estimators for time dependent convection diffusion equations, using only (SGV) stabilization or using an additional capturing term. An adaptive strategies are deduced and propped for adaptive reffnement.