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This paper proves that Scott-Vogelius finite element spaces on three-dimensional Freudenthal meshes are uniformly stable for polynomial degrees 4 and higher, resolving a long-standing conjecture for the critical cases of degrees 4 and 5. The authors develop a new mathematical framework called "barycentric skeleton-bubble calculus" to handle the complex topological constraints at vertices and construct the necessary function spaces. The result completes the theoretical foundation for using these numerical methods to solve incompressible fluid flow problems in three dimensions.
Why it matters
This work provides rigorous mathematical justification for using Scott-Vogelius finite elements in computational fluid dynamics simulations, particularly for incompressible Navier-Stokes equations. The resolution of this conjecture enables more reliable and efficient numerical methods for engineering applications including aerodynamics, blood flow modeling, and weather prediction.
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⚠️ Preprint – Noch nicht peer-reviewed
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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.