Scott-Vogelius 元在三维 Freudenthal 网格上四阶和五阶的一致稳定性：解决 Farrell-Mitchell-Scott 猜想
Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
- Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子学与生物光子学重点实验室，长春理工大学物理学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文证明了 Scott-Vogelius 有限元在三维 Freudenthal 网格上次数 k>=4 的一致 inf-sup 稳定性，解决了 Farrell-Mitchell-Scott 猜想的关键情形。
AI中文摘要：
我们建立了单位立方体上均匀 Freudenthal 四面体剖分中 Scott-Vogelius 有限元空间在多项式次数 k >= 4 时的一致 inf-sup 稳定性估计。该结果完全解决了 Farrell、Mitchell 和 Scott 关于临界次数 k = 4 和 k = 5 的第一个猜想，补充了已知的高次多项式稳定性范围。主要的数学困难源于奇异顶点处所需的复杂拓扑相容性以及相邻单元间相应的均值约束。我们通过发展统一的重心骨架-气泡演算来应对这些挑战，该演算显式构造顶点射流、边模式和面传递，以全局路由单元均值。随附的精确计算独立验证了这些有限维恒等式，并提供了可复现性数据。
英文摘要：
We establish a uniform inf-sup stability estimate for the Scott-Vogelius finite element spaces on uniform Freudenthal tetrahedralizations of the unit cube for polynomial degrees k >= 4. This result completely settles the first conjecture of Farrell, Mitchell, and Scott for the critical degrees k = 4 and k = 5, complementing the known stability range for higher polynomial degrees. The main mathematical difficulties stem from the complex topological compatibility required at the singular vertices and the corresponding mean-value constraints across adjacent elements. We tackle these challenges by developing a unified barycentric skeleton-bubble calculus that explicitly constructs vertex jets, edge modes, and face transfers to globally route element means. The accompanying exact computations independently verify these finite-dimensional identities and provide reproducibility data.
