Closure complexity of longest-edge bisection for triangular meshes

Abstract

On triangular meshes, we analyze local mesh refinement based on longest-edge bisection equipped with the serial longest-edge propagation-path closure. Ties are resolved by terminal priority: if the incoming shared edge is a longest edge of the neighboring triangle, the pair is declared terminal and that edge is bisected. For every adaptive mesh sequence $\mathcal{T}_0, \mathcal{T}_1, \ldots, \mathcal{T}_L$ with marked subset sequence $\mathcal{M}_0, \mathcal{M}_1, \ldots, \mathcal{M}_{L-1}$, we prove the cumulative closure estimate $\#\mathcal{T}_L - \#\mathcal{T}_0 \lesssim \sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell$. The proof has two ingredients. First, the finite-similarity-class theorem for planar longest-edge bisection turns the set of possible diameters into a finite union of $\sqrt{2}$-geometric lattices. Hence diameters grow by a uniform strict factor at every nonterminal step. This yields generation and spatial locality for all triangles created by a single mark. Second, a Binev--Dahmen--DeVore type charging argument converts this single-mark locality into the cumulative estimate.