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
- Rebholz, Leo
2010 Mathematics Subject Classification
- 76D05 65M60
2008 Physics and Astronomy Classification Scheme
- 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.
- SIAM J. Numer. Anal., 49 (2011) pp. 1461--1481.