Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
Abstract
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.
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BibTeX
@misc{liang2026uniform,
title = {{Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture}},
author = {Hanbing Liang and Fujun Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29608},
eprint = {2609.29608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29608},
}
RIS
TY - UNPB
TI - Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
AU - Hanbing Liang
AU - Fujun Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29608
UR - https://arxiv.org/abs/2609.29608
AB - 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.
ER -