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Interior face means form the global Crouzeix–Raviart degrees of freedom

Statement

For every interior facet $F$ , define $ϕ_{F}$ to equal $ϕ_{K, F}$ on each cell $K$ incident to $F$ and zero on all other cells, where $ϕ_{K, F}$ is the local shape polynomial of The local P1 Crouzeix–Raviart element. Then ${ϕ_{F} : F \in F^{\circ}}$ is a basis of the space in The global homogeneous Crouzeix–Raviart space. The face mean ${\overset{ˉ}{v}}_{F} : = ∣ F ∣^{- 1} \int_{F} v_{K} d S$ is single-valued for $v \in V_{h}^{C R, 0}$ and $\dim V_{h}^{C R, 0} = ∣ F^{\circ} ∣$ .

Facts & Assumptions

Given: A finite matching affine simplicial mesh, the homogeneous global space of The global homogeneous Crouzeix–Raviart space, and local unisolvence from Face averages are unisolvent for the local Crouzeix–Raviart element.

[F1]

On each cell the face-average functionals are a basis of the dual of $P_{1} (K)$ , and $ϕ_{K, F}$ has face means $δ_{F G}$ . (Face averages are unisolvent for the local Crouzeix–Raviart element)

Proof

1.1F1givenalgebra

The two traces of $v \in V_{h}^{C R, 0}$ on an interior facet have equal integrals by definition, so ${\overset{ˉ}{v}}_{F}$ is single-valued. For an interior facet $F$ , the function $ϕ_{F}$ has mean one on $F$ , mean zero on every other facet of each incident cell by [F1], and is zero on cells not incident to $F$ . Thus $ϕ_{F} \in V_{h}^{C R, 0}$ and ${\overset{ˉ}{ϕ}}_{F, G} = δ_{F G}$ for all interior facets $G$ .

2.1F1step 1.1algebra∎

Let $v \in V_{h}^{C R, 0}$ and set $w : = \sum_{F \in F^{\circ}} {\overset{ˉ}{v}}_{F} ϕ_{F}$ . On each cell, $v_{K}$ and $w_{K}$ have the same mean on every interior facet and both have mean zero on every boundary facet. Local unisolvence [F1] therefore gives $v_{K} = w_{K}$ for every cell. Hence the global functions span. Their linear independence follows by applying each single-valued face mean to a linear combination and using step 1.1. The number of basis functions is the number of interior facets, which proves the dimension formula.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Part VIII, Chapter 36, Crouzeix–Raviart approximation (standard reference, not scraped)
Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 25, Non-conforming Finite Element Methods (standard reference, not scraped)