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arXiv · 0808.2031

Piecewise polynomials on polyhedral complexes

T. McDonald·H. Schenck
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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.

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T. McDonald, H. Schenck. 2008-08-14. Piecewise polynomials on polyhedral complexes. https://arxiv.org/abs/0808.2031

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