Let Omega⊂R^3 be a domain equipped with a uniform three-dimensional Freudenthal tetrahedral mesh T_h. For every integer velocity degree k≥ 4, let V_h=[P_k(T_h)cap H^1_0(Omega)]^3 and Q_h=P_k-1^disc(T_h)cap L^2_0(Omega). The Scott–Vogelius pair (V_h,Q_h) is inf-sup stable: there exists a constant beta>0, independent of the mesh size h, such that inf_0≠ q_h∈ Q_hsup_0≠ v_h∈ V_hfrac(nabla!· v_h,q_h)_Omega|v_h|_H^1(Omega)|q_h|_L^2(Omega)≥beta.

Farrell–Mitchell–Scott Scott–Vogelius stability conjecture

Let $\Omega\subset\mathbb{R}^3$ be a domain equipped with a uniform three-dimensional Freudenthal tetrahedral mesh $\mathcal{T}_h$ . For every integer velocity degree $k\ge 4$ , let $V_h=[\mathcal{P}_k(\mathcal{T}_h)\cap H^1_0(\Omega)]^3$ and $Q_h=\mathcal{P}_{k-1}^{\mathrm{disc}}(\mathcal{T}_h)\cap L^2_0(\Omega)$ . The Scott–Vogelius pair $(V_h,Q_h)$ is inf-sup stable: there exists a constant $\beta>0$ , independent of the mesh size $h$ , such that

\inf_{0\ne q_h\in Q_h}\sup_{0\ne v_h\in V_h}\frac{(\nabla\!\cdot v_h,q_h)_{\Omega}}{\|v_h\|_{H^1(\Omega)}\|q_h\|_{L^2(\Omega)}}\ge\beta.

References

Primary source

arXiv

Additional references

The Scott-Vogelius element is inf-sup stable on Freudenthal meshes for k ≥ 4 — arXiv — Siqi Ding, Pingbing Ming, Haijun Yu, Quanfan Zhu

finite-element-methods numerical-analysis

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

An unrefereed September 2026 preprint claims to prove the conjectured stability threshold in three dimensions, but independent verification is not yet recorded.

Farrell, Mitchell, and Scott proposed that the Scott–Vogelius pair on uniform three-dimensional Freudenthal meshes is inf-sup stable for degrees $k \ge 4$ . Their 2024 paper gave numerical evidence and also formulated a separate local-kernel decomposition conjecture for $k \ge 5$ .

Known results

Stability was previously proved for degrees $k \ge 6$ , attributed to Zhang.
Farrell, Mitchell, and Scott (2024) conjectured stability for the sharp range $k \ge 4$ , based on computation rather than proof.
Their second conjecture concerns a stable local decomposition of the divergence-free kernel for $k \ge 5$ .

September 2026 claimed resolution

Siqi Ding, Pingbing Ming, Haijun Yu, and Quanfan Zhu claim uniform inf-sup stability on Freudenthal meshes for $k \ge 4$ , using explicit local-patch constructions. A separate September 3 preprint by Hanbing Liang and FuJun Liu makes the same claim for $k=4$ and $k=5$ . Both are unrefereed claims; no independent verification or refutation was found.

Current status (as of September 2026): Stability for $k \ge 6$ is established; stability for $k \ge 4$ is claimed by unrefereed preprints but remains unverified, while the separate kernel-decomposition conjecture for $k \ge 5$ remains open.

Sources

No solutions have been posted yet.