On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems

It was recently proven that, under mild restrictions, grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems converge to the Scott-Vogelius solution of that same problem. However, even though the analytical rate was only shown to be $gamma^-frac12$ (where $gamma$ is the stabilization parameter), the computational results suggest the rate may be improvable $gamma^-1$. We prove herein the analytical rate is indeed $gamma^-1$, and extend the result to other incompressible flow problems including Leray-$alpha$ and MHD. Numerical results are given that verify the theory.