  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.
