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

Abstract

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 be 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.