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 $H(\mathrm{\text{div}})$-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 $H(\mathrm{\text{div}})$-conforming correction rendering the full velocity divergence-free. For $k\ge d$, 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 $P_k\times P_0$ scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.

本文考虑了斯托克斯方程在任意形状正则单纯网格上的有限元空间 Scott-Vogelius 对的离散化。提出并分析了一种在离散 inf-sup 条件下稳定这些对的新方法。关键思想在于丰富阶连续多项式$k$Scott-Vogelius 速度空间与适当选择并明确给出的 Raviart-Thomas 气泡。这种方法的灵感来自于[X. Li 和 H. Rui，低阶无散度$H（\mathrm{\text{分区}}）$-斯托克斯流的有限元方法，IMA J. Numer。肛门。42 (2022) 3711–3734]，其中案例$k=1$被研究了。所提出的方法是压力鲁棒的，具有最佳收敛性$H^1$- 一致的速度和小的$H（\mathrm{\text{分区}}）$- 一致的校正使全速度无发散。为了$k\ge d$， 和$d$作为维度，该方法是无参数的。此外，结果表明，Raviart-Thomas 富集的附加自由度以及所有非恒定压力自由度都可以被消除，有效地导致压力鲁棒、inf-sup 稳定、最佳收敛${磷}_k\times {磷}_0$方案。讨论了实施的各个方面，数值研究证实了分析结果。