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 $k>1$.

翻译：本文针对单纯形有限元网格，引入了通用多项式阶数$k$和空间维度$d\geq2$的Crouzeix-Raviart元。我们给出了非协调基函数的显式表示，并证明了协调伴随空间（即多项式阶数为$k$的协调有限元空间）包含于Crouzeix-Raviart空间。我们证明了Crouzeix-Raviart空间可分解为协调伴随空间（的子空间）与非协调基函数张成的空间的直和。引入了与基函数双对偶的自由度，从而定义了局部逼近/插值算子。在二维情形或$k=1$时，这些自由度可分解为单纯形积分与$\left( d-1\right)$维面积分，使得在Crouzeix-Raviart函数的基表示中，所有属于网格中低维面相关基函数的系数均可由这些面积分确定。研究还表明，在更高空间维度且$k>1$时，**不存在**这样的自由度集合。