Super spline spaces of smoothnessr and degreed≥3r+2

Abstract

The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degree d with r continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super sphnes, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-B6zier method.

Stable Local Bases for Multivariate Spline Spaces

“…In particular, local bases for S r, \ d (2), where \ { =2 n&l&1 , were constructed in [11] and [4]. For general \ { , but only in the bivariate case n=2, the superspline spaces were explored in [22,28] and, more recently, in [18,19].…”

“…Moreover, the dual basis is also local and therefore provides a quasi-interpolant possessing optimal approximation order. There are well known constructions of local bases for S r d (2) in the bivariate case n=2 for all d 3r+2, see [1,21,22,26]. Stable local bases were constructed in [7,23] for some superspline subspaces, and in [17,19] for the full bivariate spline spaces S r d (2), d 3r+2.…”

On the Dimension of Bivariate Spline Space S31(Δ)

“…3.3 in [6] can be applied. Thus for n ≥ 4 our results coincide with the general results on the dimension, given in [11] and [12].…”

Smooth macro-elements on Powell-Sabin-12 splits

“…We use Bernstein-Bézier techniques as in [1,2,3,4,5,6,7,8,9,10,11,12,13,16,17]. In particular, we represent polynomials p of degree d on a triangle T := v 1 , v 2 In this paper we are interested in subspaces S of S 0 d ( ) that satisfy additional smoothness conditions.…”

“…They can also be used in situations where some of the coefficients of two different pieces of s are known. The following well-known lemma [4] (see also Lemma 3.3 of [7]) shows how this works for computing coefficients on the ring R A MDS M is called local provided that there is an integer n such that for every ξ ∈ D d, ∩ T and every triangle T in , c ξ is a linear combination of {c η } η∈Γ ξ where Γ ξ is a subset of M with Γ ξ ⊂ star n (T ). Here star n (T ) := star(star n−1 (T )) for n ≥ 2, where if U is a cluster of triangles, star(U ) is the set of all triangles that have a nonempty intersection with U .…”