The Scott-Vogelius Element is Inf-Sup Stable on Freudenthal Meshes for k ≥ 4

The Scott-Vogelius element is a classical divergence-free mixed finite element method for the Stokes problem that has been extensively studied for decades. This paper provides a rigorous mathematical proof establishing inf-sup stability on Freudenthal meshes in three dimensions for polynomial degrees k ≥ 4, extending previous results by Zhang who proved stability for k ≥ 6 in 2011. The work completes the theoretical foundation for this element, which is fundamental to computational methods in fluid dynamics and structural mechanics.