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

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Date
2010
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Journal ISSN
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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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 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

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Keywords
Navier-Stokes equations, rotational form, Scott-Vogelius element, strong mass conservation
Citation
Citation
Bowers, A. L., Cousins, B. R., Linke, A., & Rebholz, L. G. (2010). New connections between finite element formulations of the Navier-Stokes equations (Version publishedVersion, Vol. 1516). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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