The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for $k\ge 4$

Abstract

The Scott-Vogelius element is a classical divergence-free mixed finite element for the Stokes problem that has attracted decades of research attention, yet its theoretical framework remains incomplete. In two dimensions, the inf-sup stability on Freudenthal and other regular meshes has been rigorously established. In three dimensions, Zhang established inf-sup stability on Freudenthal meshes for $k\ge 6$ in 2011, while numerical evidence indicates that inf-sup stability remains true for $k=4,5$. In this paper, we strengthen Zhang's approach and prove that the Scott--Vogelius element is inf-sup stable on Freudenthal meshes for every velocity degree $k\ge 4$, thereby resolving a conjecture proposed by Farrell, Mitchell, and Scott in 2024. The proof proceeds by explicit constructions on local patches and does not rely on computer verification.

arXiv comment26 pages, 7 figures