Ce travail a porté principalement sur le développement et l'étude de méthodes numériques de type éléments finis multi-échelles pour un problème d'advection diffusion multi-échelles dominé par l'advection. Deux types d'approches sont envisagées: prendre en compte l'advection dans la construction de l'espace d'approximation, ou appliquer une méthode de stabilisation. On commence par l'étude d'un problème d'advection diffusion, dominé par l'advection, dans un milieu hétérogène. On poursuit sur des problèmes d'advection-diffusion, sous le régime où l'advection domine, posés dans un domaine perforé. On se focalise ici sur la condition aux bords de type Crouzeix Raviart pour la construction des éléments finis multi-échelles. On considère deux situations différentes selon la condition prescrite au bord des perforations: la condition de Dirichlet homogène ou la condition de Neumann homogène. Cette étude repose sur une hypothèse de coercivité.Pour finir, on se place dans un cadre général où l'opérateur d'advection-diffusion est non coercif, possiblement dominé par l'advection. On propose une approche éléments finis basée sur une mesure invariante associée à l'opérateur adjoint. Cette approche est bien posée inconditionnellement en la taille du maillage. On la compare numériquement à une méthode standard de stabilisation / This work essentially deals with the development and the study of multiscale finite element methods for multiscale advection-diffusion problems in the advection-dominated regime. Two types of approaches are investigated: Take into account the advection in the construction of the approximation space, or apply a stabilization method. We begin with advection-dominated advection-diffusion problems in heterogeneous media. We carry on with advection-dominated advection-diffusion problems posed in perforated domains.Here, we focus on the Crouzeix-Raviart type boundary condition for the construction of the multiscale finite elements. We consider two different situations depending on the condition prescribed on the boundary of the perforations: the homogeneous Dirichlet condition or the homogeneous Neumann condition. This study relies on a coercivity assumption.Lastly, we consider a general framework where the advection-diffusion operator is not coercive, possibly in the advection-dominated regime. We propose a Finite Element approach based on the use of an invariant measure associated to the adjoint operator. This approach is unconditionally well-posed in the mesh size. We compare it numerically to a standard stabilization method

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diusion partial dierential equations (PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall eciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite dierence, finite volume and continuous finite element methods for solving wave dominated problems like convection diusion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints