English abstract
The numerical approximation of convection-diffusion problems has a major diffi culty in the convection-dominated regime, where the solution typically possesses in terior and boundary layers that cannot be resolved properly. It is well-known that the classical Galerkin finite element discretization is inappropriate in this regime since it exhibits global spurious oscillations, especially in the vicinity of sharp gradients. Over the last two decades, an extensive research has been devoted to the development of methods which stabilize the classical Galerkin finite element method without losing consistency. In this thesis, the convergence and stability of some recent stabilization methods will be investigated. We first present the residual-based satbilization meth ods, and analyze the effect of introducing suitable bubble functions in the Galerkin formulation. Then, we study the local projection method and show that the fulfillment of the local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Further, the effects of the stability parameter is identified. In addition, a link between the three methods is shown and numerical tests for different types of boundary layers arising in convection diffusion problems is presented.
