Generalization to higher-dimensional and unstructured meshes

Extend the decoupling technique and completely non-diagonalizable component analysis framework to three-dimensional T-meshes supporting trivariate splines or to unstructured meshes.

Background

The results in the paper concern bivariate polynomial spline spaces over two-dimensional T-meshes. The authors explicitly identify higher-dimensional and unstructured settings as an unresolved research direction, including three-dimensional T-meshes with trivariate splines.

References

Several promising directions remain open for future research: Extension to Arbitrary and Mixed Order of Smoothness: A natural problem is to extend the decoupling framework to polynomial splines with lower or mixed orders of smoothness ($\mu < d-1$), where the continuity constraints across adjacent cells exhibit more complex algebraic couplings. Basis Function Construction: Utilizing the localized linear equations derived from the decoupled cofactors, we plan to develop constructive algorithms for stable, locally supported basis functions over general non-diagonalizable T-meshes, which is essential for isogeometric analysis (IGA) applications. Generalization to Higher Dimensions: Extending the decoupling technique and CNDC analysis framework to three-dimensional T-meshes (trivariate splines) or unstructured meshes presents an important and challenging avenue for further exploration.

— Sharp Dimension Bounds for Spline Spaces over T-meshes with Highest Order of Smoothness  (2608.19839 - Huang et al., 20 Aug 2026) in Section 5, Conclusion and future work