The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a {\em conjecture of Crouzeix-Falk} from 1989 for $p=3$. Our proof applies to {\em any odd degree} $p\ge 3$ and hence Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the piecewise polynomials of degree $p-1$ with vanishing integral form a stable Stokes pair {\em for all positive integers} $p$.

翻译：Crouzix-Raviart三角形的限定要素在Stokes方程式中稳定为$\inf$-$\sup$,任何带有至少一个内脊网格的Stokes方程式。这一结果证实了1989年Crouzix-Falk} 的假设值$p=3美元。我们的证据适用于所有正数整数的Crouzix-Raviart三角方程式3美元,因此,Crouzix-Raviart三角方程式在两个维度上稳定为$p$,而具有消失整体形式的P-1美元,所有正数整数的Crouzix-Raviart三角方形组合方块值为$p-1美元。