Uniform Poincare inequalities on vertex and cell patches

Scaling [F1] to each triangle $K$ gives $\mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2(K)}\le C{h}_K\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}$ and $\mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2(E)}\le C{h}_K^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}$ for every edge $E\subset \mathrm{\partial }K$. Shape regularity makes $h_E$ comparable to $h_K$. If $K,L$ share $E$, their traces of $v$ agree, so $\mathrm{\mid }{v}_K-v_L\mathrm{\mid }\mathrm{\mid }E{\mathrm{\mid }}^{1\mathrm{/}2}\le \mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2(E)}+\mathrm{\parallel }v-v_L{\mathrm{\parallel }}_{L^2(E)}\le C{h}_z^{1\mathrm{/}2}(\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}+\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(L)})\mathrm{.}$ Using $\mathrm{\mid }E{\mathrm{\mid }}^{1\mathrm{/}2}\ge c{h}_z^{1\mathrm{/}2}$ for cells in one shape-regular vertex star bounds the difference of neighboring means by the two gradient norms.

The star has at most $N_{\mathrm{\Gamma }}$ cells, and its dual edge graph is connected. Join each cell to a fixed cell by a graph path of length at most $N_{\mathrm{\Gamma }}-1$ and apply step 1.1 along that path. The cell Poincaré bound from step 1.1 then gives $\mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2({\omega }_z)}\le C{h}_z\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_z)}$. Since $\mathrm{\mid }{v}_{{\omega }_z}-v_K\mathrm{\mid }\le \mathrm{\mid }{\omega }_z{\mathrm{\mid }}^{-1\mathrm{/}2}\mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2({\omega }_z)}$, replacing $v_K$ by the patch mean proves the interior estimate.

For a boundary vertex patch choose one boundary edge $E_0\subset \mathrm{\partial }{\omega }_z$. By the definition of $H_0^1$ as the $H^1$-closure of compactly supported smooth functions, and the continuous local trace bound in [F1], the trace of $v$ is zero on $E_0$. The trace inequality in step 1.1 therefore gives $\mathrm{\mid }{v}_K\mathrm{\mid }\mathrm{\mid }{E}_0{\mathrm{\mid }}^{1\mathrm{/}2}=\mathrm{\parallel }v-v_K{\mathrm{\parallel }}_{L^2(E_0)}\le C{h}_z^{1\mathrm{/}2}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(K)}$ for the incident cell $K$. The same bounded-path comparison of cell means from step 2.1 controls every cell mean in the star by $C\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2({\omega }_z)}$. Summing the cell Poincaré bounds proves the boundary estimate.

For a cell patch ${\omega }_K$, the three vertex stars are edge-connected because they all contain $K$, and the number of cells is at most three times the valence bound from [F2]. Along each edge-adjacency path, neighboring cell diameters are comparable by shape regularity, so all diameters in the patch are comparable to $h_K$. The mean-zero estimate follows by applying the cell-mean comparison of step 2.1 over this connected finite graph. If $K$ meets the boundary, ${\omega }_K$ contains a boundary edge; apply the zero-trace argument of step 3.1 there and the same path comparison to all cells in the union. These two estimates are the asserted cell-patch inequalities.