Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions
<|reference_start|>Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions: 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 $k>1$.<|reference_end|>
@article{bohne2024crouzeix-raviart,
    title={Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions},
    author={Nis-Erik Bohne, Patrick Ciarlet Jr., Stefan Sauter},
    journal={arXiv preprint arXiv:2407.04361},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.04361},
    primaryClass={math.NA cs.NA}
}