Scott–Vogelius Inf–Sup Stability on Three-Dimensional Freudenthal Meshes for Polynomial Degree k ≥ 4
Year: 2026
Type: preprint
Abstract: The work addresses the conjecture of Farrell, Mitchell, and Scott concerning mesh-size-independent inf–sup stability of the Scott–Vogelius finite element pair on uniform three-dimensional Freudenthal triangulations of the unit cube. The conjectured stability range is polynomial degree (k\ge 4); previously established analytical results covered (k\ge 6). The manuscript proves the two remaining cases, (k=4) and (k=5), by constructing uniformly (H^1)-bounded right inverses of the discrete divergence. The argument combines a Bernardi–Raugel lifting of elementwise pressure means with local vertex, edge, and tetrahedral-bubble corrections. The quartic case requires two-star enlargements for a finite set of exceptional boundary vertex configurations, while the quintic case additionally uses local edge corrections. The finite-dimensional local surjectivity statements required by the proof are verified using exact rational arithmetic. The accompanying computational files provide reproducible certificates for the relevant vertex-star, two-star, and edge-star rank calculations. No floating-point rank decisions are used. Together with the previously known (k\ge 6) result, the (k=4) and (k=5) cases establish the conjectured range (k\ge 4) for the Freudenthal mesh family considered in the manuscript.
Author: Cavalcante, Claudemir
Institution: Universidade Federal do Pará
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208835804