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：Scott-Vogelius有限元对用于二维稳态Stokes方程的数值离散，是一种基于$k$次连续速度逼近与$k-1次不连续压力逼近的经典单元。该方法通过"奇异距离"（由三角形顶点$\mathbf{z}$的几何网格量$\Theta \left( \mathbf{z}\right) \geq 0$度量）并针对满足$\Theta \left( \mathbf{z}\right) =0$的顶点$\mathbf{z}$施加压力空间的局部边条件。对于任意固定正则三角剖分且$k\geq 4$时，该格式满足inf-sup稳定性。然而，当三角剖分包含近奇异顶点$0<\Theta \left( \mathbf{z}\right) \ll 1$时，inf-sup常数会恶化。本文引入一种非常简单的参数依赖型Scott-Vogelius单元修正方法，使得inf-sup常数与近奇异顶点无关。通过理论分析与数值实验表明，该修正对离散速度的无散条件影响可忽略不计。