Discrete Poincaré inequality from boundary paths

Let $\mathcal{T}$ be a finite boundary-anchored TPFA mesh and let $T_e>0$ be its interior and boundary transmissibilities. A boundary path family assigns to each cell $K$ a finite simple path ${\gamma }_K$ of adjacent interior faces, followed by one boundary face of its terminal cell; simple means it visits no cell more than once. Set ${\mathrm{\Delta }}_{e}u=u_K-u_L$ on an interior face $e=K\mathrm{\mid }L$ with a fixed orientation, and set ${\mathrm{\Delta }}_{e}u=u_K$ on a boundary face of $K$. The sign of an interior increment is reversed when the path traverses the fixed orientation backward. Put $S_K={\sum }_{e\in {\gamma }_K}{T}_e^{-1},C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}={\max }_e{\sum }_{K:e\in {\gamma }_K}\mathrm{\mid }K\mathrm{\mid }{S}_K\mathrm{.}$ Then every cell vector $u$ satisfies the discrete Poincaré inequality ${\sum }_{K\in \mathcal{K}}\mathrm{\mid }K\mathrm{\mid }{u}_K^2\le C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}{\sum }_e{T}_e({\mathrm{\Delta }}_{e}u{)}^2=C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}\mathrm{\parallel }u{\mathrm{\parallel }}_{E_h}^2\mathrm{.}$ Consequently, any uniformly regular mesh family supplied with boundary path families whose $C_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ are uniformly bounded has a mesh-uniform Poincaré constant. This theorem does not assert that every orthogonal mesh family has such a uniform path bound.