Stabilized reduced basis methods for the approximation of parametrized viscous flows

In Reduced Basis (RB) method, the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if the stable Taylor-Hood Finite Element pair is chosen. Therefore in this PhD thesis we aim to build a stabilized RB method suitable for the approximation of parametrized viscous flows. Starting from the state of the art we study the residual based stabilization techniques for parametrized viscous flows in a RB setting. We are interested in the approximation of the velocity and pressure. extit{Offline-online} computational splitting is implemented and extit{offline-only stabilization}, and extit{offline-online stabilization} are compared (as well as without a stabilization approach). Different test cases are illustrated and several classical stabilization approaches like Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square are recast into a parametric reduced order setting. The RB method is introduced as a Galerkin projection into reduced spaces, generated by basis functions chosen through a greedy (steady cases) and POD-greedy (unsteady cases) algorithms. This approach is then compared with the supremizer options to guarantee the approximation stability by increasing the corresponding parametric inf-sup condition. We also implement a rectification method to correct the consistency of extit{offline-only stabilization} approach. Several numerical results for both steady and unsteady problems are presented and compared. The goal is two-fold: to guarantee the RB inf-sup stability and to guarantee online computational savings by reducing the dimension of the online reduced basis system.