Crouzeix-Raviart triangular elements are inf-sup stable

TL;DR

This paper proves that Crouzeix-Raviart triangular finite elements are inf-sup stable for the Stokes equations across all odd degrees p ≥ 3, confirming a long-standing conjecture and extending stability results.

Contribution

It establishes the inf-sup stability of Crouzeix-Raviart elements for all odd degrees p ≥ 3, generalizing previous results and confirming a conjecture from 1989.

Findings

Proves inf-sup stability for all odd degrees p ≥ 3

Confirms the Crouzeix-Falk conjecture from 1989

Applicable to any mesh with at least one interior vertex

Abstract

The Crouzeix-Raviart triangular finite elements are $\inf$-$\sup$ stable for the Stokes equations for any mesh with at least one interior vertex. This result affirms a {\em conjecture of Crouzeix-Falk} from 1989 for $p=3$. Our proof applies to {\em any odd degree} $p\ge 3$ and hence Crouzeix-Raviart triangular finite elements of degree $p$ in two dimensions and the piecewise polynomials of degree $p-1$ with vanishing integral form a stable Stokes pair {\em for all positive integers} $p$.