Augmenting the grad-div stabilization for Taylor–Hood finite elements with a vorticity stabilization

Abstract The least squares vorticity stabilization (LSVS), proposed in N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzmán, A. Linke, and C. Merdon (“A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation,” SIAM J. Numer. Anal. , vol. 59, no. 5, pp. 2746–2774, 2021) for the Scott–Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor–Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott–Vogelius and Taylor–Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott–Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.