Local potential bases on Freudenthal meshes
Partition $\Omega = (0,1)^3$ into $N^3$ congruent cubes, where $N \ge 1$ and $h = 1/N$, and subdivide the cube with lower corner $a$ into the six tetrahedra $$a + h \operatorname{conv}\{0,\ e_{\sigma(1)},\ e_{\sigma(1)} + e_{\sigma(2)},\ (1,1,1)\}, \qquad \sigma \in S_3.$$ Call this Freudenthal mesh $\mathcal{T}_h$. For an integer $k \ge 5$ let $$\Sigma_{h,k} = \bigl\{ w \in [C^0(\overline{\Omega})]^3 : w|_T \in [\mathbb{P}_{k+1}(T)]^3\ (T \in \mathcal{T}_h),\ \operatorname{curl} w \in [C^0(\overline{\Omega})]^3 \bigr\},$$ where $\mathbb{P}_r$ denotes the polynomials of total degree at most $r$; no boundary condition is imposed. For each mesh vertex $v$, boundary vertices included, let $\omega_v$ be the union of the closed tetrahedra containing $v$. The space $\Sigma_{h,k}$ precedes the continuous Lagrange velocity space of degree $k$ in a discrete Stokes complex. Farrell, Mitchell and Scott found that a multigrid solver with vertex-star relaxation for a nearly incompressible problem on such meshes had iteration counts robust in the penalty parameter for $k \ge 5$ but not for $k < 5$, and conjectured that no basis supported on vertex stars exists for $k = 4$. Ding, Ming, Yu and Zhu proved that the Scott–Vogelius pair is inf-sup stable on Freudenthal meshes for every $k \ge 4$.
Problem. Does $\Sigma_{h,k}$, for every $N \ge 1$ and every $k \ge 5$, have a basis each of whose members is supported in some $\omega_v$? Equivalently, is $$\Sigma_{h,k} = \sum_v \{ w \in \Sigma_{h,k} : \operatorname{supp} w \subseteq \omega_v \} \qquad (N \ge 1,\ k \ge 5)?$$ A uniform bound on the norms of such decompositions is a further question, not part of this one.
Source: Farrell, Mitchell and Scott (2024), Conjecture 2(b); stated in this form, with no boundary condition, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Keywords
- AIM Open Applied Problems, finite elements, Freudenthal meshes, Stokes complex
References
@article{farrell2024,
author = {Farrell, Patrick E. and Mitchell, Lawrence and Scott, L. Ridgway},
title = {Two conjectures on the {S}tokes complex in three dimensions on {F}reudenthal meshes},
journal = {SIAM J. Sci. Comput.},
volume = {46},
number = {2},
year = {2024},
pages = {A629--A644},
doi = {10.1137/22M1533943},
eprint = {2211.05494},
archivePrefix = {arXiv},
}
@misc{ding2026,
author = {Ding, Siqi and Ming, Pingbing and Yu, Haijun and Zhu, Quanfan},
title = {The {S}cott--{V}ogelius element is inf-sup stable on {F}reudenthal meshes for $k \ge 4$},
year = {2026},
eprint = {2609.25690},
archivePrefix = {arXiv},
}
@misc{aim-freudenthal-local-potential-basis,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Local potential bases on Freudenthal meshes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/633-freudenthal-local-potential-basis.md},
note = {Entry 633 at commit a153f46, 2026-10-13},
}
Resolution claims
Local bases for finite-element potential spaces provide the algebraic decompositions needed for vertex-patch corrections. On a consistently oriented Freudenthal mesh of the cube, we prove that continuous vector polynomials of degree $k+1$ with continuous curl have a basis supported on single vertex stars for every $k\ge5$ and every mesh size. This establishes the local-support assertion in the second conjecture of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00750, "Local potential bases on Freudenthal meshes", clopen.solutions.