Piecewise polynomials on polyhedral complexes

TL;DR

This paper derives formulas for the dimension of spline spaces on polyhedral complexes, providing the first complete results for two-dimensional cases using localization and dual graph techniques.

Contribution

It introduces a novel method to compute the first three coefficients of the polynomial dimension formula for splines on polyhedral complexes, especially in 2D.

Findings

First three coefficients of the dimension polynomial are explicitly determined.

Complete formula obtained for the 2D case of polyhedral complexes.

New techniques involve localization and dual graphs for analyzing spline spaces.

Abstract

For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2 and P is simplicial, Alfeld and Schumaker determined a formula for all three coefficients of f. However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension--2 linear spaces, we obtain the first three coefficients of f(P,r,k), giving a complete answer when d=2.