Finite-volume cell-face meshes and geometric regularity

Let $d\ge 1$ and let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected open polyhedral set, meaning the interior of a finite union of bounded closed convex polytopes. A closed convex polytope is a bounded intersection of finitely many closed half-spaces with nonempty interior. A cell-face mesh $\mathcal{T}=(\mathcal{K},\mathcal{F})$ consists of finitely many nonempty cells $K=\mathrm{int}{P}_K\subset \mathrm{\Omega }$, where each $P_K$ is a closed convex polytope, with pairwise disjoint interiors and $\bar{\mathrm{\Omega }}={\bigcup }_{K\in \mathcal{K}}\bar{K}$, together with the relative interiors of the $(d-1)$-dimensional faces of their closures. Each interior face $F\in {\mathcal{F}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ is incident to exactly two cells $K,L$ and is denoted $F=K\mathrm{\mid }L$; each boundary face $F\in {\mathcal{F}}_{\mathrm{b}\mathrm{d}\mathrm{y}}$ is incident to one cell and is contained in $\mathrm{\partial }\mathrm{\Omega }$. In dimension one a face is an endpoint and has measure $\mathrm{\mid }F\mathrm{\mid }=1$.

Write $\mathrm{\mid }K\mathrm{\mid }$ for the $d$-dimensional volume of $K$, $\mathrm{\mid }F\mathrm{\mid }$ for the $(d-1)$-dimensional measure of $F$, $n_{K,F}$ for the unit normal outward from $K$, $x_K$ for a chosen point in $K$, and $x_F$ for the centroid of $F$. For every incident pair set $d_{K,F}=\mathrm{dist}(x_K,F)>0$. The cell adjacency graph has one vertex for each cell and an edge for each interior face. A mesh is boundary anchored if every cell has a finite path in this graph to a cell incident to a boundary face; such a path can be chosen simple, with no repeated cells.

For a family $({\mathcal{T}}_h)$, put $h={\max }_{K\in \mathcal{K}}\mathrm{diam}K$. The family is uniformly regular if there are fixed constants $c,C>0$ and $N_0\in \mathbb{N}$ such that, for every mesh and each of its cells and faces, $c{h}^d\le \mathrm{\mid }K\mathrm{\mid }\le C{h}^d,c{h}^{d-1}\le \mathrm{\mid }F\mathrm{\mid }\le C{h}^{d-1},ch\le d_{K,F}\le Ch,$ and each cell has at most $N_0$ faces. A family is refining if $h\to 0$. These bounds are invoked only where explicitly assumed; a cell-face mesh alone is not asserted to be uniformly regular or orthogonal.

The finite-volume convention is that a face flux $Q_{K,F}$ is signed outward from $K$. If $F=K\mathrm{\mid }L$ is an interior face, local conservation requires the single shared-flux identity $Q_{K,F}+Q_{L,F}=0$. The mesh drawing below labels the center-to-center vector and the outward normal; no orthogonality is part of this general mesh definition.

The cell-face mesh definition and the diagram fix only the objects, incidences, normals, and orientation. A two-point flux approximation adds a separate orthogonality condition; the hybrid mimetic method is formulated on the general cell-face geometry specified here.