A reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations

In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours \cite{Ali2018}. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.