Numerical analysis of projection methods for the time-dependent Navier-Stokes equations with modular grad-div stabilization

In this work, a projection method based on modular grad-div stabilization has been proposed for the evolutionary Navier-Stokes equations. The presented method can also be regarded as the improved penalty-projection method. Unlike the penalty-projection method that adds the grad-div term to the momentum equation, the proposed scheme incorporates a minor intrusive step to an existing projection code. By doing so, the proposed scheme not only holds the merits of the penalty-projection method; but also avoids solver failures and improves computational efficiency along with the increase of grad-div parameters. Furthermore, we show that the scheme is unconditionally stable and give out convergence analysis. In the final, numerical tests are done which validate the theoretical results and their efficiency.