Compatible 2D newest-vertex-bisection meshes

A compatible labeled initial mesh is a finite matching affine simplicial triangulation ${\mathcal{T}}_0$ of a polygon in ${\mathbb{R}}^2$ as in Matching affine simplicial meshes and shape regularity, equipped with labels on its edges such that each triangle has one edge labeled $0$ and two labeled $1$, and every shared edge has the same label from both incident triangles. Matching means that distinct cell interiors are disjoint and any nonempty cell intersection is a whole common face: in two dimensions it may be a vertex or an edge. For a descendant triangle of generation $g(K)$, its three edge labels, in any order, are $(g(K)+1,g(K)+1,g(K))$. The unique edge labeled $g(K)$ is its reference edge.

To bisect $K$, join the midpoint of its reference edge to the opposite vertex. The two resulting triangles have generation $g(K)+1$. In each child, the two new half-edges and the bisector receive label $g(K)+2$; the one inherited nonreference edge retains label $g(K)+1$ and is the child's reference edge. A finite bisection forest is the collection of the initial triangles and all descendants reached by finitely many such bisections; its leaves form a triangulation when all shared-edge subdivisions agree. The admissible family $\mathbb{T}({\mathcal{T}}_0)$ consists of the matching conforming leaf triangulations generated by these compatible bisections. A refinement $\mathcal{T}\le {\mathcal{T}}_{\ast }$ means every cell of ${\mathcal{T}}_{\ast }$ lies in a cell of $\mathcal{T}$.

For deterministic operational ordering, fix a finite starting mesh in this family and a total order of its cells. Enumerate those operational roots as $K_1,\dots ,K_N$ in that order; this starting mesh may already be a geometric descendant of ${\mathcal{T}}_0$, and operational numbering does not reset its generations or edge labels. Give root $K_r$ identifier $(r,\mathrm{\varnothing })$ and order its three vertices lexicographically by their two real coordinates. If a parent has identifier $(r,w)$, with $w$ a finite binary word, let $A$ be the earlier endpoint of its labeled reference edge in its stored vertex list, let $B$ be the other endpoint, let $C$ be the opposite vertex, and let $M=(A+B)\mathrm{/}2$. Its children with vertex sets $\{A,M,C\}$ and $\{B,M,C\}$ receive identifiers $(r,w0)$ and $(r,w1)$, respectively. Each child's stored vertex list is the parent list restricted to its two retained vertices, in their previous order, followed by $M$. These lists are bookkeeping orders, independent of the orientation convention for affine reference maps. Compare identifiers first by root rank, then lexicographically by binary word, with $0<1$ and a proper prefix before its extensions. This is a total order, so every finite nonempty pending set has a unique least identifier. Distinct roots or distinct binary paths have distinct identifiers, and a retained cell keeps its identifier and vertex list through all subsequent calls. This operational decoration changes no geometric bisection, reference edge, admissible triangulation or finite-element space.

The compatible initial labeling is a hypothesis, not a consequence of an arbitrary initial triangulation. This definition is two-dimensional and makes no claim about three-dimensional closure or complexity.

The geometry and closure properties of this family are separate theorems on the same page.