The Scott-Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order $k$ and a discontinuous pressure approximation of order $k-1$. It employs a "singular distance" (measured by some geometric mesh quantity $ \Theta \left( \mathbf{z}\right) \geq 0$ for triangle vertices $\mathbf{z}$) and imposes a local side condition on the pressure space associated to vertices $\mathbf{z}$ with $\Theta \left( \mathbf{z}\right) =0$. The method is inf-sup stable for any fixed regular triangulation and $k\geq 4$. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices $0<\Theta \left( \mathbf{z}\right) \ll 1$. In this paper, we introduce a very simple parameter-dependent modification of the Scott-Vogelius element such that the inf-sup constant is independent of nearly-singular vertices. We will show by analysis and also by numerical experiments that the effect on the divergence-free condition for the discrete velocity is negligibly small.

翻译：斯科特-沃格留斯有限元对是二维定常斯托克斯方程数值离散中一种常用的元，它基于多项式阶数为$k$的连续速度逼近和阶数为$k-1$的不连续压力逼近。该元采用"奇异距离"（由三角形顶点$\mathbf{z}$的某种几何网格量$\Theta \left( \mathbf{z}\right) \geq 0$度量），并对满足$\Theta \left( \mathbf{z}\right) =0$的顶点$\mathbf{z}$所关联的压力空间施加局部旁侧条件。该方法对任意固定的正则三角剖分和$k\geq 4$是inf-sup稳定的。然而，若三角剖分包含近乎奇异的顶点，即$0<\Theta \left( \mathbf{z}\right) \ll 1$，则inf-sup常数会恶化。本文引入一种非常简单的参数依赖型修改方案，使得inf-sup常数与近乎奇异顶点无关。我们通过理论分析和数值实验表明，该修改对离散速度的无散条件影响极小。