DeepAI AI Chat
Log In Sign Up

Critical Functions and Inf-Sup Stability of Crouzeix-Raviart Elements

by   C. Carstensen, et al.
Universität Zürich
Humboldt-Universität zu Berlin

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p≥5, p odd, are inf-sup stable for the Stokes problem on triangulations. For p≥4, p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula, the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability. The generalization of this theory to cover odd polynomial orders p≥5 is involved; one reason is that the macroelement classes, which have been used for even p, are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows.


page 1

page 2

page 3

page 4


On the Inf-Sup Stabillity of Crouzeix-Raviart Stokes Elements in 3D

We consider non-conforming discretizations of the stationary Stokes equa...

A CutFEM divergence–free discretization for the Stokes problem

We construct and analyze a CutFEM discretization for the Stokes problem ...

Fortin Operator for the Taylor-Hood Element

We design a Fortin operator for the lowest-order Taylor-Hood element in ...

Nonconforming finite element methods of order two and order three for the Stokes flow in three dimensions

In this study, the nonconforming finite elements of order two and order ...

Arnoldi-based orthonormal and hierarchical divergence-free polynomial basis and its applications

This paper presents a methodology to construct a divergence-free polynom...

On the Sobolev and L^p-Stability of the L^2-projection

We show stability of the L^2-projection onto Lagrange finite element spa...

Critical Points for Two-view Triangulation

Two-view triangulation is a problem of minimizing a quadratic polynomial...