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

It was recently proven in Case et al. (2010) [2] 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 (where γ is the stabilization parameter), the computational results suggest the rate may be improvable to 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.