Analysis of the Pspg Stabilization for the Evolutionary Stokes Equations Avoiding Time-step Restrictions

Optimal error estimates for the pressure stabilized Petrov–Galerkin (PSPG) method for the evolutionary Stokes equations are proved, in the case of regular solutions, without restriction on the length of the time step. These results clarify the “instability of the discrete pressure for small time steps” reported in the literature. First, the limit situation of the continuous-in-time discretization is considered and second the numerical analysis for the backward Euler scheme is carried out. The main results are applicable to higher order finite elements. The analytical results are strongly based on an appropriate approach for computing the initial velocity, which is suggested in this paper. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.