New connections between finite element formulations of the Navier--Stokes equations
- Bowers, Abigail L.
- Cousins, Benjamin R.
- Linke, Alexander
- Rebholz, Leo G.
2010 Mathematics Subject Classification
- 76D05 65M60
2008 Physics and Astronomy Classification Scheme
- 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.
- J. Comput. Phys., 229 (2010) pp. 9020--9025.