MathDB

Knupp's conjecture

Let F:[−1,1]3→R3F:[-1,1]^3\to\mathbb{R}^3 be a trilinear map defining a hexahedral finite element, and let JF(x)=det⁡DF(x)J_F(x)=\det DF(x) be its Jacobian determinant. If JF(x)>0J_F(x)>0 for every x∈∂([−1,1]3)x\in\partial([-1,1]^3), then JF(x)>0J_F(x)>0 for every x∈[−1,1]3x\in[-1,1]^3.

References

Primary source

arXiv

Additional references

computational-geometryfinite-elements
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 paper claims to prove that checking a hexahedron’s boundary is enough to certify its validity, but the result has not yet been independently confirmed.

Knupp’s conjecture asks whether validity of trilinear hexahedral finite elements can be determined from boundary behavior rather than by searching the whole volume. The supplied evidence records a new claimed proof by Paul Zhang.

September 17, 2026 claimed proof

Paul Zhang’s paper, Validating Hexahedra through their Boundaries, claims to prove Knupp’s conjecture. It also claims that the global minimum lies on the boundary and that validation reduces to finitely many quartic-root candidate points. These results would make volumetric validity testing a finite boundary computation, but the claim remains unverified.

Current status (as of September 2026): A paper claims the conjecture is solved, but independent and peer-review confirmation is absent.

Sources

Solutions 0

No solutions have been posted yet.