Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier--Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier--Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order $\mathcal O(h^2)$ in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. We also consider the analysis of the fully discrete case with the backward Euler method as time integrator.