Develop consistent numerical integration for limit 1- and 2-form subdivision functions

Develop quadrature and truncation strategies that enable direct numerical integration of limit Loop-subdivision basis functions for 1-form and 2-form subdivision schemes, ensuring that the truncation is performed consistently across 0-, 1-, and 2-form spaces so that de Rham complex compatibility is maintained.

Background

The authors discuss a future direction of using smooth limit subdivision functions directly in simulations to enable higher-regularity discretizations, but note that numerical integration becomes challenging because Loop limit functions are represented by infinite sequences of polynomial patches densely clustered near extraordinary vertices.

Although quadrature methods with truncation exist for scalar limit functions, it is unclear how to extend these to the 1- and 2-form subdivision spaces introduced in the paper, and how to coordinate truncation across k to preserve compatibility.

References

There exist approaches to integrate such functions using quadrature and a truncation of the sequence of polynomial patches, see for example . However, it is not clear how these ideas extend to our 1- and 2-form subdivision scheme and how the truncation can be done consistently across the k-form spaces to maintain their compatibility.

— Subdivision $k$-Form Spaces within the Finite Element Exterior Calculus Framework  (2604.02015 - Piel et al., 2 Apr 2026) in Section 6 Conclusion