In the TrioCFD code, the discretization of the Navier-Stokes equations leads to a three-step algorithm (see Chorin'67, Temam'68): velocity prediction, pressure solution, velocity correction. If an implicit time discretization scheme is to be used, the pressure solution step is particularly costly. Thus, most simulations are performed using an explicit time scheme, for which the time step depends on the mesh size, which can be very restrictive. We would like to develop an implicit time discretization scheme using a stabilized formulation of the Navier-Stokes problem based on explicit T-coercivity (see Ciarlet-Jamelot'25). It would then be possible to solve an implicit scheme directly without a correction step, which could significantly improve the performance of the calculations. This would also allow the use of the P1-P0 finite element pair, which is frugal in terms of degrees of freedom but unstable for a classical formulation.