Inf-sup stabilized Scott-Vogelius pairs on general shape-regular simplicial grids by Raviart-Thomas enrichment

This paper considers the discretization of the Stokes equations with Scott-Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order k of the Scott-Vogelius velocity space with appropriately chosen and explicitly given Raviart-Thomas bubbles. This approach is inspired by [X. Li and H. Rui, A low-order divergence-free H(div)-conforming finite element method for Stokes flows, IMA J. Numer. Anal.42 (2022) 3711-3734], where the case k=1 was studied. The proposed method is pressure-robust, with optimally converging H-1-conforming velocity and a small H(div)-conforming correction rendering the full velocity divergence-free. For k >= d, with d being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart-Thomas enrichment and also all non-constant pressure degrees of freedom can be eliminated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent P(k )x P(0 )x P-0 scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.