Dimension of mixed splines on polytopal cells

Abstract

The dimension of planar splines on polygonal subdivisions of degree at most d is known to be a degree two polynomial for d 0. For planar C r splines on triangulations this formula is due to Alfeld and Schumaker; the formulas for planar splines with varying smoothness conditions across edges on convex polygonal subdvisions are due to Geramita, McDonald, and Schenck. In this paper we give a bound on how large d must be for the known polynomial formulas to give the correct dimension of the spline space. Bounds are given for central polytopal complexes in three dimensions, or polytopal cells, with varying smoothness across two-dimensional faces. In the case of tetrahedral cells with uniform smoothness r we show that the known polynomials give the correct dimension for d ≥ 3r + 2; previously Hong and separately Ibrahim and Schumaker had shown that this bound holds for planar triangulations. All bounds are derived using techniques from computational commutative algebra.

A Lower Bound for Splines on Tetrahedral Vertex Stars

“…Put J 1 = x r+1 , y r+1 : (x + y) r+1 ; I 1 can be obtained from J 1 by the change of coordinates x → x , y → −x + y . In [13,Lemma 7.18] it is shown that the initial ideal in(J 1 ) with respect to lexicographic order consists of the dim(J 1 ) d lexicographically largest monomials in the variables x and y. In other words, in(J 1 ) in lexicographic order is a so-called lex segment ideal (see [24,Chapter 2]).…”

“…It turns out that I 1 is a complete intersection generated in degrees (r + 1)/2 , (r + 1)/2 . This implies that the Hilbert function of I 1 has the following form (for a proof see [13,Corollary 7.17]):…”

Counting the dimension of splines of mixed smoothness: A general recipe, and its application to meshes of arbitrary topologies

“…for quite small m relative to r and it is possible to obtain quite accurate estimates for the smallest such m. However, this equation typically holds in degree far lower than dimension formulas are actually known (see [7]), so we focus on giving some coarse estimates that are easy to derive. Proposition 4.14.…”

“…Computing the dimension of spline spaces is a highly non-trivial task in general for splines in more than one variable. Initiated by Strang [19,20], this is by now a classical topic in approximation theory and has been studied in a wide range of planar settings; e.g., on triangulations, polygonal meshes, and T-meshes [18,2,4,17,16,11,7,21,12,22]. Nonpolynomial spline spaces have also been studied in the same vein; e.g., [5].…”

A lower bound for splines on tetrahedral vertex stars

“…Otherwise we call the vertex star an open vertex star. In Equations 15 and 16 of [1], Alfeld, Neamtu, and Schumaker define functions (in terms of simple geometric and combinatorial data of ∆) which we denote by LB (∆, d, r) (11) and LB (∆, d, r) (12), respectively. In [1,Theorem 3], it is shown that dim H r d (∆) = LB (∆, d, r) for d ≥ 3r + 2 if ∆ is a closed vertex star and that dim H r d (∆) = LB (∆, d, r) for d ≥ 3r + 2 if ∆ is an open vertex star.…”