Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
Year: 2026
Type: preprint
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 arise from the topological compatibility conditions at singular vertices and the corresponding mean-value constraints across adjacent elements. We develop a unified barycentric skeleton-bubble calculus that explicitly constructs vertex jets, edge modes, and face transfers to globally route element means.The main theorem constructs, for every fixed k ≥ 4, a uniformly bounded linear right inverse of the discrete divergence operator, with a bound independent of the mesh size. The proof combines stable element-mean and vertex lifts, supported edge-star lifts in degrees four and five, explicit quartic and quintic mean-transfer constructions, and a low-degree element-bubble closure.Exact-arithmetic computational certificates and deterministic verification scripts accompany the analytic proof and independently verify the finite-dimensional identities used in the construction.
Source: arXiv (Cornell University)
Authors: Liang, Hanbing, Liu, Fujun
Institution: Changchun University of Science and Technology
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207550244