We discuss two types of discrete inf-sup conditions for the Taylor-Hood family $Q_k$-$Q_{k-1}$ for all $k\in \mathbb{N}$ with $k\ge 2$ in 2D and 3D. While in 2D all results hold for a general class of hexahedral meshes, the results in 3D are restricted to meshes of parallelepipeds. The analysis is based on an element-wise technique as opposed to the widely used macroelement technique. This leads to inf-sup conditions on each element of the subdivision as well as to inf-sup conditions on the whole computational domain.

翻译：我们讨论了二维和三维中所有$k\in \mathbb{N}$且$k\ge 2$的Taylor-Hood族$Q_k$-$Q_{k-1}$的两种离散inf-sup条件。在二维情形下，所有结果对于一般类六面体网格成立，而在三维情形下，结果限于平行六面体网格。该分析基于逐单元技术，而非广泛使用的宏单元技术。这既导出了剖分中每个单元上的inf-sup条件，也导出了整个计算域上的inf-sup条件。