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

TL;DR

This paper analyzes discrete inf-sup conditions for the Taylor-Hood finite element family across 2D and 3D meshes, introducing an element-wise approach that broadens understanding of stability conditions in computational fluid dynamics.

Contribution

It presents a novel element-wise analysis of inf-sup conditions for Taylor-Hood elements, extending applicability to general 2D meshes and specific 3D parallelepiped meshes.

Findings

Inf-sup conditions established for all k ≥ 2 in 2D and 3D.

Results hold for general 2D hexahedral meshes.

3D results are restricted to parallelepiped meshes.

Abstract

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.