Boundedness and divergence commutation of RT0 interpolation

Let $a_F$ be the vertex opposite facet $F$ in $K$ and define ${\varphi }_{K,F}(x)=(x-a_F)\mathrm{/}(d\mathrm{\mid }K\mathrm{\mid })$. On $F$, $(x-a_F)\cdot n_K$ is the altitude $H_F$, so ${\int }_F{\varphi }_{K,F}\cdot n_K=H_F\mathrm{\mid }F\mathrm{\mid }\mathrm{/}(d\mathrm{\mid }K\mathrm{\mid })=1$ by the simplex pyramid identity $d\mathrm{\mid }K\mathrm{\mid }=H_F\mathrm{\mid }F\mathrm{\mid }$. On any other facet, $a_F$ and $x$ lie in its hyperplane, hence ${\varphi }_{K,F}\cdot n_K=0$. Thus $I_{K}v={\sum }_{F\subset \mathrm{\partial }K}{m}_{K,F}(v){\varphi }_{K,F}$, where $m_{K,F}$ is the outward moment. By [F2], [F7], and Cauchy–Schwarz, $\mathrm{\mid }{m}_{K,F}(v)\mathrm{\mid }\le C(h_K^{(d-2)\mathrm{/}2}\mathrm{\parallel }v{\mathrm{\parallel }}_K+h_K^{d\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_K)$. Also [F3] gives $\mathrm{\parallel }{\varphi }_{K,F}{\mathrm{\parallel }}_K\le C{h}_K\mathrm{\mid }K{\mathrm{\mid }}^{-1\mathrm{/}2}\le C{h}_K^{1-d\mathrm{/}2}$. Multiplying and summing the fixed $d+1$ terms proves $\mathrm{\parallel }{I}_{K}v{\mathrm{\parallel }}_K\le C(\mathrm{\parallel }v{\mathrm{\parallel }}_K+h_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_K)$. Squaring and summing cells yields the global stability estimate since $h_K\le h$.

The divergence of $I_{K}v\in {\mathrm{RT}}_0(K)$ is constant. For smooth-neighborhood approximants to $v{\mathrm{\mid }}_K$ from [F5], Gauss–Green [F4] says the sum of their outward facet moments equals the integral of their divergence. The trace bound [F2] and H1 convergence let both sides pass to the limit, so ${\int }_K\mathrm{div}{I}_{K}v={\sum }_{F\subset \mathrm{\partial }K}{m}_{K,F}(v)={\int }_K\mathrm{div}v$. Dividing by $\mathrm{\mid }K\mathrm{\mid }$ gives the stated cell average.

If $v\in H_0^1(D;{\mathbb{R}}^d)$, choose compactly supported smooth vector fields converging to it in H1 by [F5]. Their boundary facet moments are zero; the trace bound [F2] passes this zero moment to $v$. Hence $I_{h}v$ has zero boundary facet flux, and [F1] gives membership in $H_0(\mathrm{div};D)$; [F6] then gives $I_{h}v\in V_h$.