We design a Fortin operator for the lowest-order Taylor-Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $P_2$-$P_0$ and the augmented Taylor-Hood element in 3D.

翻译：我们设计了一个Fortin操作器,用于任何层面的泰勒-胡德最低级元素,该元素以前仅建于2D。 在施工中,我们为差异校正操作器使用相近边缘泡沫功能。这自然导致一种可替代的内积稳定减少的有限元素配对。此外,我们提供了一种反向实例,说明内积稳定性,因此存在一个Fortin操作器,用于2美元-P_0美元和3D中增加的泰勒-胡德元素。