WIAS Preprint No. 1516, (2010)

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


  • Bowers, Abigail L.
  • Cousins, Benjamin R.
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Rebholz, Leo G.

2010 Mathematics Subject Classification

  • 76D05 65M60

2008 Physics and Astronomy Classification Scheme

  • 47.11.Fg


  • Navier-Stokes equations, rotational form, Scott-Vogelius element, strong mass conservation


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.

Appeared in

  • J. Comput. Phys., 229 (2010) pp. 9020--9025.

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