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Closure complexity of longest-edge bisection for triangular meshes

  • Yuwen Li
  • Zhiyuan Yang
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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 𝒯0, 𝒯1, …, 𝒯L with a sequence of marked sets β„³0, β„³1, …, β„³Lβˆ’1, we prove the cumulative closure estimate #𝒯Lβˆ’#𝒯0 β‰²βˆ‘β„“=0Lβˆ’1#β„³β„“. The proof derives single-mark locality from the uniform multiplicative gap between possible descendant diameters implied by finite similarity classes, and converts this locality into the cumulative estimate through a Binev--Dahmen--DeVore-type charging argument.

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Cite

BibTeXRIS
BibTeX
@misc{li2026closure,
  title = {{Closure complexity of longest-edge bisection for triangular meshes}},
  author = {Yuwen Li and Zhiyuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04003},
  eprint = {2609.04003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04003},
}
RIS
TY  - UNPB
TI  - Closure complexity of longest-edge bisection for triangular meshes
AU  - Yuwen Li
AU  - Zhiyuan Yang
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04003
UR  - https://arxiv.org/abs/2609.04003
AB  - 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 𝒯_0, 𝒯_1, …, 𝒯_L with a sequence of marked sets β„³_0, β„³_1, …, β„³_Lβˆ’1, we prove the cumulative closure estimate #𝒯_Lβˆ’#𝒯_0 β‰²βˆ‘_β„“=0^Lβˆ’1#β„³_β„“. The proof derives single-mark locality from the uniform multiplicative gap between possible descendant diameters implied by finite similarity classes, and converts this locality into the cumulative estimate through a Binev--Dahmen--DeVore-type charging argument.
ER  -