Fortin Operator for the Taylor-Hood Element

TL;DR

This paper constructs a Fortin operator for the lowest-order Taylor-Hood element in any dimension, extending previous 2D results, and introduces an alternative stable finite element pair, while also providing counterexamples for other elements.

Contribution

It presents a novel construction of a Fortin operator for the Taylor-Hood element in arbitrary dimensions and introduces an alternative inf-sup stable finite element pair.

Findings

Fortin operator constructed for 3D Taylor-Hood element

Counterexample showing instability of certain finite element pairs in 3D

Alternative stable finite element pair proposed

Abstract

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.