The local P1 Crouzeix–Raviart element

Let $d\ge 2$ and let $K$ be a nondegenerate affine $d$-simplex with vertices $a_0,\dots ,a_d$ and facets $F_i$ opposite $a_i$. Write ${\lambda }_0,\dots ,{\lambda }_d$ for its barycentric coordinate functions, so ${\lambda }_i(a_j)={\delta }_{ij}$ and ${\sum }_i{\lambda }_i=1$. The local Crouzeix–Raviart polynomial space is ${\mathbb{P}}_1(K)$, the real affine functions on $K$. Its face-average functionals are ${\sigma }_{K,F_i}(p):=\frac{1}{\mathrm{\mid }{F}_i\mathrm{\mid }}{\int }_{F_i}pdS,p\in {\mathbb{P}}_1(K)\mathrm{.}$ The candidate dual shape polynomial associated with $F_i$ is ${\varphi }_{K,F_i}:=1-d{\lambda }_i\mathrm{.}$ The next item proves that these $d+1$ functionals are unisolvent and that the displayed polynomials form their dual basis. No pointwise continuity across different cells is part of this local element.