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We discuss two types of discrete inf-sup conditions for the Taylor-Hoodfamily $Q_k$-$Q_{k-1}$ 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, theresults in 3D are restricted to meshes of parallelepipeds. The analysis isbased on an element-wise technique as opposed to the widely used macroelementtechnique. This leads to inf-sup conditions on each element of the subdivisionas well as to inf-sup conditions on the whole computational domain.

A short note on inf-sup conditions for the Taylor-Hood family $Q_k$-$Q_{k-1}$