Crouzeix-Raviart triangular elements are inf-sup stable

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 conjecture of Crouzeix-Falk from 1989 for p = 3. Our proof applies to any odd degree p >= 3 and concludes the overall stability analysis: 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 for all positive integers p.