WIAS Preprint No. 1510, (2010)

Improving mass conservation in FE approximations of the Navier Stokes equations using continuous velocity fields: A connection between grad-div stabilization and Scott--Vogelius elements


  • Case, Michael
  • Ervin, Vincent
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Rebholz, Leo

2010 Mathematics Subject Classification

  • 76D05 65M60

2008 Physics and Astronomy Classification Scheme

  • 47.11.Fg


  • incompressible Navier-Stokes equations, mixed finite elements, stabilized finite elements, grad-div stabilization, Taylor-Hood element, Scott-Vogelius element


This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. Under mild restrictions, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that, again under these mild restrictions, the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, in this setting, we provide theoretical justification that choosing the parameter large does not destroy the solution. A limiting result is also proven for the general case. Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations.

Appeared in

  • SIAM J. Numer. Anal., 49 (2011) pp. 1461--1481.

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