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PreprintarXiv licencearXiv

The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for k ≥ 4

  • Siqi Ding
  • Pingbing Ming
  • Haijun Yu
  • Quanfan Zhu
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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≥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≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{ding2026scottvogelius,
  title = {{The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for $k \geq 4$}},
  author = {Siqi Ding and Pingbing Ming and Haijun Yu and Quanfan Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25690},
  eprint = {2609.25690},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25690},
}
RIS
TY  - UNPB
TI  - The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for k ≥ 4
AU  - Siqi Ding
AU  - Pingbing Ming
AU  - Haijun Yu
AU  - Quanfan Zhu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25690
UR  - https://arxiv.org/abs/2609.25690
AB  - 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≥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≥ 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.
ER  -