A CONNECTION BETWEEN SCOTT-VOGELIUS AND GRAD-DIV STABILIZED TAYLOR-HOOD FE APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS

作者：Case Michael A^*; Ervin Vincent J; Linke Alexander; Rebholz Leo G

来源：SIAM Journal on Numerical Analysis, 2011, 49(4): 1461-1481.

DOI：10.1137/100794250

• Navier-Stokes equations
• finite element method
• Scott-Vogelius elements
• grad-div stabilization
• strong mass conservation

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. With a particular mesh construction, 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 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, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. 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.

• 出版日期2011