三角形网格最长边二分法的闭包复杂度
Closure complexity of longest-edge bisection for triangular meshes
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中文总结 AI 辅助
本文研究三角形网格最长边二分法的闭包复杂度，结合终端优先级解决平局，证明了自适应网格序列的累积闭包估计，其证明基于有限相似类的乘性间隙与Binev等的充电论证。
中文摘要 AI 辅助
针对三角形网格，本文分析基于最长边二分法结合串行最长边传播路径闭包的局部网格细化，平局由终端优先级解决：若传入共享边是相邻三角形的最长边，则该对被声明为终端并对该边二分。对每个带标记集序列$\u039c_0,\u039c_1,\u039c_{L-1}$的自适应网格序列$\u03a4_0,\u03a4_1,\u03a4_L$，本文证明累积闭包估计$\u0023\u03a4_L-\u0023\u03a4_0 \u2272 \u2211_{\u2113=0}^{L-1}\u0023\u039c_\u2113$。该证明从有限相似类隐含的可能后代直径间的均匀乘性间隙导出单标记局部性，再通过Binev--Dahmen--DeVore型充电论证将此局部性转化为累积估计。
英文摘要
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 a sequence of marked sets $\mathcal{M}_0, \mathcal{M}_1, \ldots, \mathcal{M}_{L-1}$, we prove the cumulative closure estimate $$\#\mathcal{T}_L-\#\mathcal{T}_0 \lesssim\sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell.$$ The proof 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.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
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