Short Note: New connections between finite element formulations of the Navier-Stokes equations

Authors:
Abigail L. Bowers;Benjamin R. Cousins;Alexander Linke;Leo G. Rebholz
Affiliations:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States;Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States;Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany;Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States
Venue:
Journal of Computational Physics
Year:
2010

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Abstract

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 be 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.