Crouzeix-Raviart triangular elements are inf-sup stable

Abstract

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