Affine simplicial meshes and RT/BDM spaces

Assume Countable Choice. Let $d\in \{2,3\}$ and let $D\subset {\mathbb{R}}^d$ be a bounded, connected polyhedral Lipschitz domain. A matching affine simplicial mesh and its shape regularity have the exact meaning of Matching affine simplicial meshes and shape regularity; additionally, call the mesh facet-connected when its dual graph (one vertex per cell and one edge per shared $(d-1)$-facet) is connected. Write $h={\max }_{K\in {\mathcal{T}}_h}\mathrm{diam}K$.

For $k\ge 0$, ${\mathbb{P}}_k(K)$ is the real scalar polynomial space of total degree at most $k$, and ${\tilde{\mathbb{P}}}_k$ is the real homogeneous scalar polynomial space of degree exactly $k$ (for $k=0$, the constants). Define ${\mathrm{RT}}_k(K)={\mathbb{P}}_k(K;{\mathbb{R}}^d)+x{\tilde{\mathbb{P}}}_k(K),{\mathrm{BDM}}_k(K)={\mathbb{P}}_k(K;{\mathbb{R}}^d)(k\ge 1)\mathrm{.}$ For either local family $X_k$, its global H(div)-conforming space is

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The zero-normal RT0 flux space and mean-zero P0 pressure space are $V_h:={\mathrm{RT}}_0({\mathcal{T}}_h)\cap H_0(\mathrm{div};D),Q_h:=\{q\in L^2(D):q{\mathrm{\mid }}_K\text{ is constant for every }K,{\int }_{D}q=0\}\mathrm{.}$ For each mesh facet $F$, fix one unit normal $n_F$; on a cell boundary use the outward unit normal $n_K$, which equals either $n_F$ or $-n_F$. The notation ${\mathbb{P}}_0({\mathcal{T}}_h)$ means cellwise constants. No existence of a mesh for every $D$ is asserted by this definition.