A characterization of supersmoothness of multivariate splines

We consider spline functions over simplicial meshes in $\mathbb{R}^n$. We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as \emph{supersmoothness}, which plays a role in the construction of multivariate splines and in the finite element method. In this paper we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.


Introduction
Polynomial splines over a simplicial partition of a domain in R n (a triangular mesh in 2D, a tetrahedral mesh in 3D, and so on) are functions whose pieces are polynomials up to a certain degree d and which join together with some order of continuity r. Such spline functions may have extra orders of smoothness at a vertex of the mesh, a property known as supersmoothness as suggested by Sorokina [13]. For example, the Clough-Tocher macroelement, which is C 1 piecewise cubic, is twice differentiable at the refinement point, as first observed by Farin [5], and so this element can be said to have supersmoothness of order 2 at that point.
For the construction of splines or finite elements with higher orders of continuity, it is important to recognize and make use of supersmoothness. For example, it plays a role in many of the macroelement constructions surveyed by Lai and Schumaker [8], where applications of splines to approximation theory and computer-aided geometric design are discussed. The concept of supersmoothness is also relevant to the finite element method. Motivated by structure-preserving or compatible discretizations there has recently been an increased interest in investigating the use of splines for vector fields and differential complexes [2,3,4,7]. The de Rham complex reveals a connection between smooth, e.g., C 1 , finite elements and the Stokes problem in fluid mechanics. In a discrete de Rham complex, the spline spaces for the velocity field may inherit the supersmoothness of the scalar field, [2,4,7,11]. Thus, supersmoothness is also of importance in the study of these problems.
Since Farin's observation about the Clough-Tocher element, Sorokina, in [13] and [14] has derived further supersmoothness properties of polynomial splines, and in particular higher order supersmoothness in a cell in 2D; see equation (4). More recently, Shekhtman and Sorokina [12] observed that supersmoothness is a phenomenon of more general spline functions, not only piecewise polynomials. Their results imply that at the vertex of a triangulation with m incoming edges all having different slopes, any C r spline with r ≥ m − 2 has supersmoothness, i.e., the spline has supersmoothness of order at least r + 1. This holds as long as the spline pieces themselves have C r+1 continuity.
The results of [12] were the motivation for this paper. If we simplify the framework of [12] and assume that all the spline pieces are C ∞ smooth, which is the case for polynomials and many other functions of interest, can we extend the results to higher orders of supersmoothness and also to higher Euclidean space dimensions? Our solution is to simplify the problem by deriving a characterization of supersmoothness in terms of the degeneracy of polynomial spline spaces over the cell (in Theorem 1). Using this, the maximal order of supersmoothness at a vertex can be determined once a general formula for the dimensions of the polynomial spline spaces over the cell is known. At the end of the paper we apply these results to various cell configurations.

Cells and supersmoothness
We start with some definitions.

Cells
Suppose v is an interior vertex of a simplicial partition of R n , n ≥ 2. We call the collection ∆ of n-simplices T in the mesh that share the common vertex v a cell and we let Ω = ∪ T ∈∆ T . For example, in 2D a cell is a sequence of triangles ∆ = {T 1 , T 2 , . . . , T m }, m ≥ 3, that form a star-shaped polygon Ω, as in Figure 1. In the special case that m = 3, ∆ is known as a Clough-Tocher split since it can also be constructed by refinement. We could start with any triangle T in the plane (the outer triangle in the figure), then let v be any point inside T and connect the three edges of T to v, thus creating three sub-triangles of T . In 3D a cell is a collection of tetrahedra. A simple example is the Alfeld split, constructed by choosing a tetrahedron T , then any point v inside T and connecting v to the four triangular faces of T . The resulting cell has four tetrahedra, as in Figure 2.

Splines
We want to study functions that are defined piecewise over the simplices of ∆. Let r be an integer, r ≥ 0. Then we let S r (∆) = {s ∈ C r (Ω) : s| T ∈ C ∞ , T ∈ ∆}, and we will call a function s ∈ S r (∆) a spline. We are assuming here that for each T ∈ ∆ there is some open set T ⊂ R n containing T and a function s T ∈ C ∞ (T ) such that s(x) = s T (x) for x ∈ T . The pieces s T could, for example, be polynomials of any degree (in which case we can take T = R n ), or rational functions, trigonometric functions, and so on.

Supersmoothness
Now we look at enhanced smoothness of splines at v. We will say that a spline s ∈ S r (∆) has supersmoothness of order ρ ≥ r at v if all its pieces s| T , T ∈ ∆, have common derivatives up to order ρ at the point v. In this case we will follow convention and write s ∈ C ρ (v) even though s will not in general be C ρ in any neighbourhood of v. To make this clearer let us consider the following two cases.
• If a spline s ∈ S r (∆) is in C r+1 (v) then s has all derivatives of order r + 1 at v, but not in general in any neighbourhood of v.
• If a spline s ∈ S r (∆) is in C ρ (v) for some ρ ≥ r + 2 then s will not in general even have derivatives of order ρ at v because such derivatives are only defined if s has derivatives of order ρ−1 in a neighbourhood of v. However, the restriction of s to any straight line in Ω passing through v will have smoothness of order ρ.

Taylor approximations
Our aim is to characterize supersmoothness in terms of the degeneracy of polynomial splines. The first step in the derivation is to study Taylor approximations. Let f be a function in C ∞ (B) for some domain B ⊂ R n and let α = (α 1 , . . . , α n ) be a multi-index, with α 1 , . . . , α n ≥ 0. We denote the corresponding partial derivative of f by Then, with respect to a point v = (v 1 , . . . , v n ) ∈ B, we denote the Taylor approximation of f of order ρ ≥ 0 by where |α| = α 1 + · · · + α n , α! = α 1 ! · · · α n !.
We will make use of the following property of these Taylor approximations.

Lemma 1
Let v, w be distinct points in R n and let e = [v, w] be the line segment connecting them. Let B ⊂ R n be some domain containing e. Suppose that f, g ∈ C ∞ (B) and that f | e = g| e . Then, for any Proof. We can represent the line segment e parametrically as Since f and g are equal on e, we also have and so We want to generalize this property to derivatives of f and g. To do this we first show Then, for any integer ρ ≥ 0 and any multi-index β with |β| ≤ ρ,

From Lemmas 1 and 2 we obtain
Lemma 3 Let v, w, e, B be as in Lemma 1. Suppose that f, g ∈ C ∞ (B) and that for some Then, for any ρ ≥ 0, Proof. If |β| > ρ, equation (2) trivially holds since both sides are equal to 0. If |β| ≤ ρ, by Lemma 2, equation (2) is equivalent to and by Lemma 1, this is implied by equation (1). 2

Characterization of supersmoothness
We are now approaching our characterization of supersmoothness.

Polynomial spline spaces and degeneracy
For integers r and d with 0 ≤ r ≤ d let where Π d is the linear space of polynomials in R n of degree at most d. Thus S r d (∆) is the usual space of polynomial splines on ∆ of smoothness r and degree at most d.
By definition, for any cell ∆ and any r ≥ 0 we have Π d ⊂ S r d (∆). Sometimes, however, depending on ∆ and r, we might have S r d (∆) = Π d . In this case S r d (∆) contains no 'true' splines, only polynomials, and we view S r d (∆) as being 'degenerate' in this sense.
As an example, the space S r r (∆) is degenerate for any r ≥ 0 and any cell ∆.

Piecewise Taylor approximations
Next recall the more general spline space S r (∆), r ≥ 0, and let s ∈ S r (∆). For any ρ ≥ 0, we can make a piecewise Taylor approximation of s by piecing together the individual Taylor approximations of order ρ at v of the pieces of s. We will refer to this piecewise approximation as T ∆,ρ s. Due to Lemma 3 we next show Lemma 4 If s ∈ S r (∆) for any r ≥ 0 then for any ρ ≥ 0, T ∆,ρ s ∈ S r ρ (∆).
Proof. We only need to show that T ∆,ρ s ∈ C r (Ω). Let T 1 , T 2 ∈ ∆ be two simplices that share a common (n − 1)-dimensional face F . The face F is the union of all the line segments e that connect v to the (n − 2)-dimensional face of F opposite to v. The pieces s| T 1 and s| T 2 have the same derivatives up to order r on e. Therefore, by Lemma 3, the two Taylor approximations T v,ρ (s| T 1 ) and T v,ρ (s| T 2 ) have the same derivatives up to order r on e. Therefore, they have the same derivatives up to order r on the whole face F . Thus T ∆,ρ s ∈ C r (Ω) as claimed.

Characterization
Finally we arrive at the characterization. .
Proof. Suppose that S r ρ (∆) is degenerate and let s ∈ S r (∆). Since S r ρ (∆) = Π ρ , Lemma 4 implies that T ∆,ρ s ∈ Π ρ . It then follows that T ∆,ρ s ∈ C ρ (v). For each T ∈ ∆, the two functions s| T and T v,ρ (s| T ) have the same derivatives at v up to order ρ. Therefore, also s ∈ C ρ (v). This proves that S r (∆) ∈ C ρ (v).

Maximal order of supersmoothness
We can also consider the mos (maximal order of supersmoothness) of S r (∆), i.e., To characterize this, observe that we have a nested sequence of spaces, Thus, for any cell ∆ and any r ≥ 0, there is a unique highest degree d ≥ r such that S r d (∆) is degenerate. From Theorem 1 we deduce Corollary 1 mos S r (∆) = max{d ≥ r : S r d (∆) is degenerate}.

Applications
We now apply the characterization theorem to some concrete examples. For a cell ∆ in R n and smoothness r ≥ 0 the spline space For some cell configurations degeneracy is known for specific degrees d > r. We then conclude from Theorem 1 that all splines in S r (∆) have supersmoothness of order d, but we do not know whether d is optimal. However, if we know the dimensions of all the spaces S r d (∆), d > r, we obtain the maximal supersmoothness from Corollary 1 by finding the largest d satisfying (3).
We note also that Alfeld [1] has computed the dimension of many spline spaces over various kinds of cell. These computational results also determine supersmoothness by Theorem 1 or Corollary 1.

Clough-Tocher split
In R 2 , when ∆ has three triangles it is a Clough-Tocher split, ∆ CT , and, using the theory of Bernstein-Bézier polynomials, Farin showed in [5, Theorem 7] that S r r+1 (∆ CT ) is degenerate for any r ≥ 1. He then concluded in [5, We can now apply Theorem 1 to conclude more generally that S r (∆ CT ) ∈ C r+1 (v) for r ≥ 1. However, this is not the optimal supersmoothness for general r.

An arbitrary cell in 2D
Sorokina made a substantial generalization of Farin's result. She showed in [13, Theorem 3.1] that if ∆ has m triangles, and the m interior edges have different slopes, then for 0 ≤ r ≤ d, The proof was based on comparing the dimension of S r d (∆) with those of superspline spaces. Since ρ in (4) is independent of the degree d, one might expect a more general result. This was also suggested by the work of Shekhtman and Sorokina [12]. From (4) it follows that there is at least one order of supersmoothness when r ≥ m − 2 for any degree d ≥ r. Shekhtman and Sorokina showed that this is also true for more general splines, in other words, in our notation, S r (∆) ∈ C r+1 (v) when r ≥ m − 2. Their proof was based on expressing partial derivatives as linear combinations of directional derivatives along the edges meeting at v. Using Corollary 1, we can now improve this result to match that of the polynomial case, in other words, we can remove the 'd' in (4). To do this, we first transform the dimension formula of Lai and Schumaker [8] into a more suitable form.
Using the fact that we can rewrite (6) as Then, using the fact that (−x) + + x = x + , the result follows.
Proof. By Corollary 1, it is sufficient to determine the highest degree d ≥ r such that S r d (∆) is degenerate, i.e., such that dim S r d (∆) = dim Π d . To do this we use Lemma 5. Suppose m v < m. If d = r + 1, the second term in (5) is strictly positive and so S r r+1 (∆) is non-degenerate. Therefore S r d (∆) is degenerate if and only if d = r. Otherwise, m v = m. Then considering the third term in (5), S r d (∆) is degenerate if and only if τ v,j ≤ 0 for all j = 1, . . . , d − r, or equivalently τ v,d−r ≤ 0, which is equivalent to As an example, for the Clough-Tocher split we have m = m v = 3 and so mos S r (∆ CT ) = r + r + 1 2 . (8)

The Alfeld split in R n
The dimensions of the spaces S r d (∆) are not currently known for a general cell ∆ in R n for n ≥ 3. However, they are known in special cases. One of these is the Alfeld split in R n . In R n , n ≥ 2, the split is constructed by choosing any n-dimensional simplex T and splitting it into n+1 smaller simplices by choosing an arbitrary interior point v in T and connecting it to each of the n + 1 faces (of dimension n − 1) of T . We denote this split by ∆ A,n . The 3D case ∆ A,3 is shown in Figure 2.

Theorem 3
The maximal order of supersmoothness of the Alfeld split is mos S r (∆ A,n ) = ρ n,r .
We note that Theorem 3 in the case n = 2 agrees with the supersmoothness of the Clough-Tocher split in equation (8).

The ∆ k,n split
Worsey and Farin [15] proposed an alternative generalization of the Clough-Tocher split to R n , using recursion through the Euclidean dimensions. To split an n-simplex T , they first split the faces of T of dimension 2 (triangles) by making a Clough-Tocher split. They next split each 3-face (a tetrahedron) F of T by choosing any point in the relative interior of F and connecting it to the twelve triangles on the boundary of F constructed in the previous step. They continue in a similar way, next splitting faces of T of dimension 4 and so on. Part of a Worsey-Farin split in 3D is shown in Figure 3, viewed as a refinement of an Alfeld split. One of the subsimplices of the Alfeld split has been split into three. Let us consider a more general split. We choose any Euclidean dimension k, 1 ≤ k ≤ n. We then initialize the splitting by splitting each k-face F of T by choosing any point in the relative interior of F and connecting it to the (k − 1)-faces of F . Then, for j = k + 1, . . . , n in sequence, we split each j-face F of T by choosing any point in the relative interior of F and connecting it to the (j + 1) × j! k! = (j + 1)! k! simplices of dimension (j − 1) on the boundary of F constructed in the previous step. The resulting split of T is a cell around the point v in the interior of T chosen at the last step (j = n). It has (n + 1)!/k! sub-simplices and we denote it by ∆ k,n .
By construction, each of the (n − 1)-faces of T is itself split into a ∆ k,n−1 split. A split ∆ k,n , k < n, can also be viewed as a refinement of a split ∆ k+1,n .
It was shown by Worsey and Farin [15] that S 1 2 (∆ 2,n ) is degenerate for any n ≥ 2. Based on this observation, they concluded, as 'an interesting aside', that their C 1 piecewise-cubic element has C 2 supersmoothness at v. Theorem 1 implies more generally that S 1 (∆ 2,n ) ∈ C 2 (v). Using now degeneracy over the Alfeld split in R k we obtain a more general result.

Theorem 4
The maximal order of supersmoothness of a ∆ k,n split, 2 ≤ k ≤ n, is bounded as follows: ρ k,r ≤ mos S r (∆ k,n ) ≤ ρ n,r .
Proof. First let r ≤ d ≤ ρ k,r . We will show that S r d (∆ k,n ) is degenerate. The proof of this is similar to that of [15,Theorem 3.2] and is by induction on n ≥ k. Consider first n = k. Since ∆ k,k is a k-dimensional Alfeld split it follows from Lemma 5 that S r d (∆ k,k ) is degenerate. Now suppose n > k and let s ∈ S r d (∆ k,n ). Let F be one of the (n − 1)-faces of T . Let w be the point in the relative interior of F used to make the (n − 1)-dimensional split ∆ k,n−1 (F ) of F in the construction of ∆ k,n . Let be any point on the line segment (v, w] and let F p be the (n − 1)-simplex which passes through p and is parallel to F . The split ∆ k,n−1 (F ) induces an analogous split ∆ k,n−1 (F p ). By the induction hypothesis, S r d (∆ k,n−1 (F p )) is degenerate and so all the pieces of s meeting at [v, w] have common derivatives within F p up to order d at p. Since all these pieces join continuously along [v, w], they also have common derivatives along [v, w]. Therefore all these pieces are the same polynomial and thus s belongs to S r d (∆ A,n ). Since d ≤ ρ k,r ≤ ρ n,r , it follows, as in the proof of Theorem 3, that s ∈ Π d .
This proves the lower bound on mos S r (∆ WF,n ). To prove the upper bound we just need to observe that ∆ k,n is a refinement of an Alfeld split ∆ A,n , which implies that 5.5 The ∆ n−1,n split Consider the special case of the ∆ n−1,n split, which has n(n + 1) subsimplices. It can be constructed by first making an Alfeld split ∆ A,n (= ∆ n,n ) of an n-simplex T using some interior point v. We then choose an interior point of each boundary face F (an (n − 1)-simplex) of T and use it to split F into n subsimplices and then connect them to v. Let us say that ∆ n−1,n is aligned if, for every face F , the splitting point chosen for F is the unique point in F that is collinear with v and the vertex of T opposite F . This is what Schenck and Sorokina [10] called a facet split.
The largest possible d in both cases gives the result by Corollary 1. 2 It is remarked in [10,Remark 4.3] that for r = 1, the dimension formula (9) also holds even without the collinearity condition, from which we conclude that for an arbitrary ∆ n−1,n split, mos S 1 (∆ n−1,n ) = ρ n−1,1 = n − 1.

2-cells
Finally, we consider a slightly different kind of cell, constructed as follows. Let T be an n-dimensional simplex and choose an interior point v of T and connect it to just one (n − 1)-face of T , forming a simplex T 1 contained in T . We now let T 2 be the the closure of T \ T 1 . The two elements T 1 and T 2 form what we will call a 2-cell, ∆ 2 = {T 1 , T 2 }. Of course it is not a cell of simplices because T 2 is not a simplex. Figure 4 shows a 2-cell in 2D. Now we can consider the supersmoothness of splines in S r (∆ 2 ). Even though 2-cells do not occur in simplicial meshes, the local configuration of edges emanating from v could occur in a polytopal mesh if we allowed non-convex polytopes. Shektman and Sorokina [12] studied this kind of configuration in 2D and showed that the order of supersmoothness is at least r + 1 for any r ≥ 0. We can now extend this result using our characterization. Even though a 2-cell contains the non-simplicial element T 2 , the intersection of T 1 and T 2 is the union of n faces (of dimension (n − 1)) and so our characterization of supersmoothness at v also holds for a 2-cell, i.e., we can apply Theorem 1 and Corollary 1 to a 2-cell ∆ 2 . To use these results we need the dimensions of the spline spaces S r d (∆ 2 ), 0 ≤ r ≤ d. Lemma 6 For any 0 ≤ r ≤ d, dim S r d (∆ 2 ) = dim Π d + dim Π d−n(r+1) . Proof. We have dim S r d (∆ 2 ) = dim Π d + dim S 0 , where S 0 = {s ∈ S r d (∆ 2 ) : s ≡ 0 on T 2 }. Letting F 1 , . . . , F n be the (n − 1)-dimensional faces common to T 1 and T 2 , we have S 0 = {p ∈ Π d : D α p| F i = 0, |α| ≤ r, i = 1, . . . , n}.