论文标题
Inf-Sup稳定的Scott - Vogelius对Raviart的一般简单网格 - Thomas Enrichment
Inf-sup stabilized Scott--Vogelius pairs on general simplicial grids by Raviart--Thomas enrichment
论文作者
论文摘要
本文认为，在任意形状的调节简单网格上，用Scott对Stokes方程的离散化。提出和分析了一种稳定这些对离散INF-SUP条件的新型方法。关键思想是丰富了斯科特（Scott）$ k $的连续多项式 - vogelius速度空间，并适当选择并明确给予了raviart- thomas bubbles。这种方法的灵感来自[Li/Rui，Ima J. Numer。肛门，2021]，研究了$ k = 1 $的情况。所提出的方法是压力刺激性的，具有最佳收敛$ \ boldsymbol {h}^1 $ - 合并速度和一个小的$ \ boldsymbol {h}（\ mathrm {div}）$ - 顺应校正，呈现完整的速度差异。对于$ k \ ge d $，$ d $是尺寸，该方法无参数。此外，可以表明，可以凝结raviart的额外自由度 - thomas富集以及所有非恒定压力自由度，有效地导致压力稳定，iff-sup稳定，最佳收敛性$ \ boldsymbol {p} p} _k \ times p_0 p_0 $ $ $ $。讨论了实施的各个方面，数值研究证实了分析结果。
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 [Li/Rui, IMA J. Numer. Anal, 2021], where the case $k=1$ was studied. The proposed method is pressure-robust, with optimally converging $\boldsymbol{H}^1$-conforming velocity and a small $\boldsymbol{H}(\mathrm{div})$-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 condensated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent $\boldsymbol{P}_k \times P_0$ scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.
