Extend Powell–Sabin and T-spline methods to compatible k-form spaces on irregular meshes

Construct finite element spaces for differential k-forms on irregular (e.g., general polygonal) meshes based on Powell–Sabin splines or T-splines that preserve compatibility in the sense of forming a discrete de Rham complex. Specifically, generalize these scalar (0-form) spline families to k-forms in a way that maintains commuting differentiation and exactness properties analogous to Finite Element Exterior Calculus.

Background

The paper surveys existing spline generalizations for irregular meshes, noting that Powell–Sabin splines and T-splines work effectively for scalar functions (0-forms) but currently lack compatible extensions to k-forms. Compatibility here refers to the preservation of de Rham complex structure, critical for stability and structure preservation in PDE discretizations.

The authors’ framework targets subdivision-based k-form spaces that preserve the de Rham complex, contrasting with the absence of compatible k-form constructions for other spline families on irregular meshes. Addressing this gap would broaden the toolbox for structure-preserving methods beyond subdivision-based approaches.