The inf-sup constant for $hp$-Crouzeix-Raviart triangular elements

TL;DR

This paper establishes a lower bound for the inf-sup constant in the discretization of the 2D stationary Stokes equation using Crouzeix-Raviart elements, showing it depends logarithmically on polynomial degree and is mesh-independent.

Contribution

It provides a mesh-independent lower bound for the inf-sup constant for $hp$-Crouzeix-Raviart elements, with mild mesh assumptions and polynomial degree dependence.

Findings

Inf-sup constant bounded below independently of mesh size

Dependence on polynomial degree is logarithmic

Applicable to a wide class of triangulations

Abstract

In this paper, we consider the discretization of the two-dimensional stationary Stokes equation by Crouzeix-Raviart elements for the velocity of polynomial order $k\geq1$ on conforming triangulations and discontinuous pressure approximations of order $k-1$. We will bound the inf-sup constant from below independent of the mesh size and show that it depends only logarithmically on $k$. Our assumptions on the mesh are very mild: for odd $k$ we require that the triangulations contain at least one inner vertex while for even $k$ we assume that the triangulations consist of more than a single triangle.