Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions

TL;DR

This paper introduces generalized Crouzeix-Raviart finite element spaces on simplicial meshes in multiple dimensions, providing explicit basis representations, decomposition properties, and degrees of freedom analysis for non-conforming elements.

Contribution

It extends Crouzeix-Raviart elements to arbitrary polynomial order and dimensions, with explicit basis functions, decomposition, and degrees of freedom characterization.

Findings

Conformal space is contained in the Crouzeix-Raviart space.

Decomposition of the space into conforming and non-conforming parts.

Facet-based degrees of freedom are possible in 2D and for k=1, but not in higher dimensions or for higher order k.

Abstract

In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$.