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 γ1⁄2 (where γ is the stabilization parameter), the computational results suggest the rate may be improvable γ-1. We prove herein the analytical rate is indeed γ -1, and extend the result to other incompressible flow problems including Leray-α and MHD. Numerical results are given that verify the theory