A posteriori error estimates for the time dependent Navier-Stokes system coupled with the convection-diffusion-reaction ; Estimations d'erreur a posteriori pour le couplage des équations de Navier-Stokes avec l'équation de convection-diffusion-réaction

The exact solution of partial differential equations is generally difficult to calculate. For this reason, numerical methods have been developed, which allow to find approximate solutions. In this thesis, we are interested in the unsteady incompressible Navier-Stokes equations which describe the behaviour of a fluid. We choose a case where the force applied to the fluid as well as its viscosity depend on another variable, for example its temperature or the concentration of a certain material in the fluid. So we need to couple the Navier-Stokes equations with the convection-diffusion-reaction equation which describes the transport of the additional variable with the fluid velocity. Our study was developed along three lines. The problem is discretized using the Euler scheme for the time discretization and the "P1-bubble/P1/P1" finite elements for the space discretization. In a first step, we study the error estimate of the coupled problem assuming that the diffusion coefficient in the transport equation is a constant. This estimate involves two types of error indicators, the first related to the discretization in time and the second to the discretization in space. In a second step, we take up the study of the coupled problem, assuming this time that the diffusion coefficient also depends on the transported variable. We start by showing the existence and the conditional uniqueness of the solution of this new problem. Then we establish the error estimate of the latter by following the same steps as those carried out in our first study. Finally, numerical simulations are elaborated using FreeFem++ software to validate the theoretical results. The time and space adaptive simulations demonstrate the usefulness of the approach. ; La solution exacte des équations aux dérivées partielles est en général difficile à calculer. Pour cela, des méthodes numériques ont été développées, qui nous permettent de trouver des solutions approchées. Dans cette thèse, nous nous intéressons aux équations de Navier-Stokes incompressibles ...