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

Abstract

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 [Formula: see text] 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 [Formula: see text]-conforming finite element method for Stokes flows, IMA J. Numer. Anal. 42 (2022) 3711–3734], where the case [Formula: see text] was studied. The proposed method is pressure-robust, with optimally converging [Formula: see text]-conforming velocity and a small [Formula: see text]-conforming correction rendering the full velocity divergence-free. For [Formula: see text], with [Formula: see text] 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 [Formula: see text] scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.