The global homogeneous Crouzeix–Raviart space

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded connected Lipschitz polyhedral domain with a finite matching affine simplicial mesh $\mathcal{T}$. Here Lipschitz polyhedral means that the boundary is a finite union of flat simplex faces and, near each boundary point after a rigid coordinate change, $\mathrm{\Omega }$ lies on one side of a Lipschitz graph. Let ${\mathcal{F}}^{\circ }$ and ${\mathcal{F}}^{\mathrm{\partial }}$ be its interior and boundary facets. For a cellwise affine function $v$, write $v_K$ for its polynomial restriction to $K$. The broken gradient ${\mathrm{\nabla }}_{h}v$ is defined by $({\mathrm{\nabla }}_{h}v){\mathrm{\mid }}_K:=\mathrm{\nabla }{v}_K$. For a facet $F$ and an integrable function $w$ on $F$, let $h_F:=\mathrm{diam}(F)$ and write ${\mathrm{avg}}_F(w):=\mathrm{\mid }F{\mathrm{\mid }}^{-1}{\int }_{F}wdS$. The homogeneous Crouzeix–Raviart space is $V_h^{CR,0}:=\{v\in L^2(\mathrm{\Omega }):v_K\in {\mathbb{P}}_1(K)\text{ for each }K\in \mathcal{T},{\int }_F(v_{K^{+}}-v_{K^{-}})dS=0\text{ for }F=K^{+}\cap K^{-}\in {\mathcal{F}}^{\circ },{\int }_F{v}_{K}dS=0\text{ for }F\in {\mathcal{F}}^{\mathrm{\partial }}\}\mathrm{.}$ The ordering $K^{+},K^{-}$ on an interior facet is arbitrary; the zero-mean condition is independent of that ordering. The functions need not agree pointwise on a facet and need not belong to $H^1(\mathrm{\Omega })$.