Finite-volume families with a regular simplicial companion

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz polyhedral domain, $d\ge 1$. A regular simplicial companion family consists of refining uniformly regular boundary-anchored cell-face meshes in Finite-volume cell-face meshes and geometric regularity, and, for each mesh, a matching simplicial subdivision ${\mathcal{S}}_h$ of its cell closures with the following finite geometric data and uniform constants $c,C,J>0$.

Each cell closure $\bar{K}$ is the union of at most $J$ closed nondegenerate $d$-simplices with pairwise disjoint interiors. Each simplex has diameter at most $Ch$ and inradius at least $ch$, and the chosen cell point $x_K$ is a subdivision vertex. The subdivision of each cell closure is a fan from $x_K$ over a triangulation of its boundary, and subdivisions agree on shared faces. Each original face closure $\bar{F}$ is a union of at most $J$ closed subdivision facets. In dimension one these mean the two subintervals from $x_K$ to the endpoints.

For each subdivision vertex $z$, let $\mathcal{K}(z)$ consist of the original cells whose closures contain $z$. This set has at most $J$ elements. For any $K,L\in \mathcal{K}(z)$ there is a path of at most $J$ cells in $\mathcal{K}(z)$, from $K$ to $L$, consecutive cells sharing an original face whose closure contains $z$. If $z\in \mathrm{\partial }\mathrm{\Omega }$, every $K\in \mathcal{K}(z)$ has such a path to a cell with a boundary face whose closure contains $z$. These are connectivity checks on finite vertex stars, not an assumption of an analytic inequality.

For shared face values $v_F$, cell values $v_K$, and homogeneous boundary values $v_F=0$, define the piecewise constant state ${\mathrm{\Pi }}_{h}v{\mathrm{\mid }}_K=v_K$ and $D_{h,p}(v{)}^p={\sum }_K{\sum }_{F\subset \mathrm{\partial }K}\mathrm{\mid }F\mathrm{\mid }{d}_{K,F}^{1-p}\mathrm{\mid }{v}_F-v_K{\mathrm{\mid }}^p(1<p<\mathrm{\infty })\mathrm{.}$ The mean interpolator $I_{h}v$ has cell and face values equal to the corresponding means of a Sobolev function, with face means interpreted by the simplex trace; for $v\in W_0^{1,p}(\mathrm{\Omega })$ its boundary face means are zero. Define $E_{h}v$ to be continuous and affine on each subdivision simplex, with nodal value $\mathrm{\mid }\mathcal{K}(z){\mathrm{\mid }}^{-1}{\sum }_{K\in \mathcal{K}(z)}{v}_K$ at an interior vertex $z$ and zero at a boundary vertex. Thus $E_h$ is an explicit linear function of the cell values; its definition does not assume its stability.

These conditions allow nonorthogonal cells. A nonempty example in every dimension is the uniform cubical partition of $(0,1{)}^d$, with cell centers, a fixed matching barycentric subdivision of every face, and the cones from cell centers to those face simplices. Scaling leaves finitely many simplex shapes and bounded vertex stars; coordinate adjacency connects each interior star and connects each boundary star to a boundary face. In dimension one use equal subintervals and their midpoints. Hence all the displayed constants can be chosen independently of refinement.