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≥2 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 degrees of feedom can be split into simplex and d-1 dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which correspond 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 not exist in higher space dimension and k>1.