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- ItemOn the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Linke, Alexander; Rebholz, Leo G.; Wilson, Nicholas E.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.
- ItemNew connections between finite element formulations of the Navier-Stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bowers, Abigail L.; Cousins, Benjamin R.; Linke, Alexander; Rebholz, Leo G.We show the velocity solutions to the convective, skew-symmetric, and rotational Galerkin finite element formulations of the Navier-Stokes equations are identical if Scott-Vogelius elements are used, and thus all three formulations will the same pointwise divergence free solution velocity. A connection is then established between the formulations for grad-div stabilized Taylor-Hood elements: under mild restrictions, the formulations' velocity solutions converge to each other (and to the Scott-Vogelius solution) as the stabilization parameter tends to infinity. Thus the benefits of using Scott-Vogelius elements can be obtained with the less expensive Taylor-Hood elements, and moreover the benefits of all the formulations can be retained if the rotational formulation is used. Numerical examples are provided that confirm the theory