Polynomial splines over a simplicial partition of a domain in (a triangular mesh in 2D, a tetrahedral mesh in 3D, and so on) are functions whose pieces are polynomials up to a certain degree and which join together with some order of continuity . Such spline functions may have extra orders of smoothness at a vertex of the mesh, a property known as supersmoothness as suggested by Sorokina . For example, the Clough-Tocher macroelement, which is piecewise cubic, is twice differentiable at the refinement point, as first observed by Farin , 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
, 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., , 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  and  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  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 incoming edges all having different slopes, any spline with has supersmoothness, i.e., the spline has supersmoothness of order at least . This holds as long as the spline pieces themselves have continuity.
The results of  were the motivation for this paper. If we simplify the framework of  and assume that all the spline pieces are 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.
2 Cells and supersmoothness
We start with some definitions.
Suppose is an interior vertex of a simplicial partition of , . We call the collection of -simplices in the mesh that share the common vertex a cell and we let . For example, in 2D a cell is a sequence of triangles , , that form a star-shaped polygon , as in Figure 1.
In the special case that , is known as a Clough-Tocher split since it can also be constructed by refinement. We could start with any triangle in the plane (the outer triangle in the figure), then let be any point inside and connect the three edges of to , thus creating three sub-triangles of .
In 3D a cell is a collection of tetrahedra. A simple example is the Alfeld split, constructed by choosing a tetrahedron , then any point inside and connecting to the four triangular faces of . The resulting cell has four tetrahedra, as in Figure 2.
We want to study functions that are defined piecewise over the simplices of . Let be an integer, . Then we let
and we will call a function a spline. We are assuming here that for each there is some open set containing and a function such that for . The pieces could, for example, be polynomials of any degree (in which case we can take ), or rational functions, trigonometric functions, and so on.
Now we look at enhanced smoothness of splines at . We will say that a spline has supersmoothness of order at if all its pieces , , have common derivatives up to order at the point . In this case we will follow convention and write even though will not in general be in any neighbourhood of . To make this clearer let us consider the following two cases.
If a spline is in then has all derivatives of order at , but not in general in any neighbourhood of .
If a spline is in for some then will not in general even have derivatives of order at because such derivatives are only defined if has derivatives of order in a neighbourhood of . However, the restriction of to any straight line in passing through will have smoothness of order .
3 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 be a function in for some domain and let be a multi-index, with . We denote the corresponding partial derivative of by
Then, with respect to a point , we denote the Taylor approximation of of order by
We will make use of the following property of these Taylor approximations.
Let be distinct points in and let be the line segment connecting them. Let be some domain containing . Suppose that and that . Then, for any , .
Proof. We can represent the line segment parametrically as
where . Letting for some , we find that
Since and are equal on , we also have
We want to generalize this property to derivatives of and . To do this we first show
Let and suppose for some domain containing . Then, for any integer and any multi-index with ,
Proof. From the definition of , for ,
Let be as in Lemma 1. Suppose that and that for some ,
Then, for any ,
4 Characterization of supersmoothness
We are now approaching our characterization of supersmoothness.
4.1 Polynomial spline spaces and degeneracy
For integers and with let
where is the linear space of polynomials in of degree at most . Thus is the usual space of polynomial splines on of smoothness and degree at most .
By definition, for any cell and any we have . Sometimes, however, depending on and , we might have . In this case contains no ‘true’ splines, only polynomials, and we view as being ‘degenerate’ in this sense.
We will say that is degenerate if .
As an example, the space is degenerate for any and any cell .
4.2 Piecewise Taylor approximations
Next recall the more general spline space , , and let . For any , we can make a piecewise Taylor approximation of by piecing together the individual Taylor approximations of order at of the pieces of . We will refer to this piecewise approximation as . Due to Lemma 3 we next show
If for any then for any , .
Proof. We only need to show that . Let be two simplices that share a common -dimensional face . The face is the union of all the line segments that connect to the -dimensional face of opposite to . The pieces and have the same derivatives up to order on . Therefore, by Lemma 3, the two Taylor approximations and have the same derivatives up to order on . Therefore, they have the same derivatives up to order on the whole face . Thus as claimed.
Finally we arrive at the characterization. For , we will write if for all .
For , if and only if is degenerate.
Proof. Suppose that is degenerate and let . Since , Lemma 4 implies that . It then follows that . For each , the two functions and have the same derivatives at up to order . Therefore, also . This proves that .
Conversely, suppose that . Then since , every polynomial spline is in and therefore . This shows that is degenerate.
4.4 Maximal order of supersmoothness
We can also consider the mos (maximal order of supersmoothness) of , i.e.,
To characterize this, observe that we have a nested sequence of spaces,
Therefore, if is non-degenerate for some , then is non-degenerate for all . Thus, for any cell and any , there is a unique highest degree such that is degenerate. From Theorem 1 we deduce
We now apply the characterization theorem to some concrete examples. For a cell in and smoothness the spline space , with , is degenerate if
For some cell configurations degeneracy is known for specific degrees . We then conclude from Theorem 1 that all splines in have supersmoothness of order , but we do not know whether is optimal. However, if we know the dimensions of all the spaces , , we obtain the maximal supersmoothness from Corollary 1 by finding the largest satisfying (3).
5.1 Clough-Tocher split
In , when has three triangles it is a Clough-Tocher split, , and, using the theory of Bernstein-Bézier polynomials, Farin showed in [5, Theorem 7] that is degenerate for any . He then concluded in [5, Corollary 8] that for any .
We can now apply Theorem 1 to conclude more generally that for . However, this is not the optimal supersmoothness for general .
5.2 An arbitrary cell in 2D
Sorokina made a substantial generalization of Farin’s result. She showed in [13, Theorem 3.1] that if has triangles, and the interior edges have different slopes, then for ,
The proof was based on comparing the dimension of with those of superspline spaces. Since in (4) is independent of the degree , one might expect a more general result. This was also suggested by the work of Shekhtman and Sorokina . From (4) it follows that there is at least one order of supersmoothness when for any degree . Shekhtman and Sorokina showed that this is also true for more general splines, in other words, in our notation, when . Their proof was based on expressing partial derivatives as linear combinations of directional derivatives along the edges meeting at . 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  into a more suitable form.
Suppose has triangles and suppose there are different slopes among the interior edges of . For ,
where , and if and otherwise.
Suppose has triangles and suppose there are different slopes among the interior edges of . Then for ,
Proof. By Corollary 1, it is sufficient to determine the highest degree such that is degenerate, i.e., such that . To do this we use Lemma 5. Suppose . If , the second term in (5) is strictly positive and so is non-degenerate. Therefore is degenerate if and only if . Otherwise, . Then considering the third term in (5), is degenerate if and only if for all , or equivalently , which is equivalent to
As an example, for the Clough-Tocher split we have and so
5.3 The Alfeld split in
The dimensions of the spaces are not currently known for a general cell in for . However, they are known in special cases. One of these is the Alfeld split in . In , , the split is constructed by choosing any -dimensional simplex and splitting it into smaller simplices by choosing an arbitrary interior point in and connecting it to each of the faces (of dimension ) of . We denote this split by . The 3D case is shown in Figure 2.
Using the theory of Bernstein-Bézier polynomials, Worsey and Farin showed in [15, Lemma 3.1] that is degenerate. From this, Theorem 1 implies that . But we can make a further generalization by invoking the recently derived dimension formula of Foucart and Sorokina  and Schenck . Let us define, for and ,
The maximal order of supersmoothness of the Alfeld split is
Therefore, is degenerate if and only if , or equivalently if
By Corollary 1, the maximal order of supersmoothness is the largest such , i.e.,
or equivalently, .
For example, , and in particular, , which shows that the macro-element on the Alfeld split described in [8, Section 18.3] has supersmoothness.
5.4 The split
Worsey and Farin  proposed an alternative generalization of the Clough-Tocher split to , using recursion through the Euclidean dimensions. To split an -simplex , they first split the faces of of dimension 2 (triangles) by making a Clough-Tocher split. They next split each 3-face (a tetrahedron) of by choosing any point in the relative interior of and connecting it to the twelve triangles on the boundary of constructed in the previous step. They continue in a similar way, next splitting faces of 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 , . We then initialize the splitting by splitting each -face of by choosing any point in the relative interior of and connecting it to the -faces of . Then, for in sequence, we split each -face of by choosing any point in the relative interior of and connecting it to the
simplices of dimension on the boundary of constructed in the previous step. The resulting split of is a cell around the point in the interior of chosen at the last step (). It has sub-simplices and we denote it by .
For example, in 2D, is a Clough-Tocher split and is a Powell-Sabin 6-split. In 3D, is an Alfeld split, is a Worsey-Farin split and is a Worsey-Piper split.
By construction, each of the -faces of is itself split into a split. A split , , can also be viewed as a refinement of a split .
It was shown by Worsey and Farin  that is degenerate for any . Based on this observation, they concluded, as ‘an interesting aside’, that their piecewise-cubic element has supersmoothness at . Theorem 1 implies more generally that . Using now degeneracy over the Alfeld split in we obtain a more general result.
The maximal order of supersmoothness of a split, , is bounded as follows:
Proof. First let . We will show that is degenerate. The proof of this is similar to that of [15, Theorem 3.2] and is by induction on . Consider first . Since is a -dimensional Alfeld split it follows from Lemma 5 that is degenerate. Now suppose and let . Let be one of the -faces of . Let be the point in the relative interior of used to make the -dimensional split of in the construction of . Let
be any point on the line segment and let be the -simplex
which passes through and is parallel to . The split induces an analogous split . By the induction hypothesis, is degenerate and so all the pieces of meeting at have common derivatives within up to order at . Since all these pieces join continuously along , they also have common derivatives along . Therefore all these pieces are the same polynomial and thus belongs to . Since , it follows, as in the proof of Theorem 3, that .
This proves the lower bound on . To prove the upper bound we just need to observe that is a refinement of an Alfeld split , which implies that
for any . Thus if is non-degenerate, so is is non-degenerate,
5.5 The split
Consider the special case of the split, which has subsimplices. It can be constructed by first making an Alfeld split () of an -simplex using some interior point . We then choose an interior point of each boundary face (an -simplex) of and use it to split into subsimplices and then connect them to .
Let us say that is aligned if, for every face , the splitting point chosen for is the unique point in that is collinear with and the vertex of opposite . This is what Schenck and Sorokina  called a facet split.
The maximal order of supersmoothness of an aligned split is
Proof. The dimensions of the polynomial spline spaces for an aligned split were derived by Schenck and Sorokina . For ,
where is as in Theorem 3 and
Therefore, is degenerate if and only if . Since when , this is equivalent to the condition that , which holds when
The largest possible in both cases gives the result by Corollary 1.
Finally, we consider a slightly different kind of cell, constructed as follows. Let be an -dimensional simplex and choose an interior point of and connect it to just one -face of , forming a simplex contained in . We now let be the the closure of . The two elements and form what we will call a 2-cell, . Of course it is not a cell of simplices because is not a simplex. Figure 4 shows a 2-cell in .
Now we can consider the supersmoothness of splines in . Even though 2-cells do not occur in simplicial meshes, the local configuration of edges emanating from could occur in a polytopal mesh if we allowed non-convex polytopes. Shektman and Sorokina  studied this kind of configuration in 2D and showed that the order of supersmoothness is at least for any . We can now extend this result using our characterization. Even though a 2-cell contains the non-simplicial element , the intersection of and is the union of faces (of dimension ) and so our characterization of supersmoothness at also holds for a -cell, i.e., we can apply Theorem 1 and Corollary 1 to a -cell . To use these results we need the dimensions of the spline spaces , .
For any ,
Proof. We have
Letting be the -dimensional faces common to and , we have
Let be any equation for the face , . Then we can express any uniquely in the form
where if and if .
For , .
For example, in , and in 3D, .
-  Peter Alfeld, MDS, http://www.math.utah.edu/~alfeld/MDS/.
-  Peter Alfeld and Tatyana Sorokina, Linear differential operators on bivariate spline spaces and spline vector fields, BIT Numerical Mathematics 56 (2016), no. 1, 15–32.
-  Douglas N Arnold, Richard S Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta numerica 15 (2006), 1–155.
-  Snorre H Christiansen and Kaibo Hu, Generalized finite element systems for smooth differential forms and Stokes’ problem, Numerische Mathematik (2018), 1–45.
-  Gerald Farin, Bézier polynomials over triangles and the construction of piecewise polynomials, Report TR/91. Uxbridge, UK: Department of Mathematics, Brunel University (1980), 1–66.
-  Simon Foucart and Tatyana Sorokina, Generating dimension formulas for multivariate splines, Albanian Journal of Mathematics 7 (2013), no. 1, 24–35.
-  Guosheng Fu, Johnny Guzman, and Michael Neilan, Exact smooth piecewise polynomial sequences on Alfeld splits, arXiv preprint arXiv:1807.05883 (2018), 1–28.
-  Ming-Jun Lai and Larry L Schumaker, Spline functions on triangulations, vol. 110, Cambridge University Press, 2007.
-  Hal Schenck, Splines on the Alfeld split of a simplex and type A root systems, Journal of Approximation Theory 182 (2014), 1–6.
-  Hal Schenck and Tatyana Sorokina, Subdivision and spline spaces, Constructive Approximation 47 (2018), no. 2, 237–247.
-  Larry L Schumaker, On super splines and finite elements, SIAM journal on numerical analysis 26 (1989), no. 4, 997–1005.
-  Boris Shektman and Tatyana Sorokina, Intrinsic supersmoothness, Journal of Concrete and Applicable Mathematics 13 (2015), 232–241.
-  Tatyana Sorokina, Intrinsic supersmoothness of multivariate splines, Numerische Mathematik 116 (2010), no. 3, 421–434.
-  , Redundancy of smoothness conditions and supersmoothness of bivariate splines, IMA Journal of Numerical Analysis 34 (2014), no. 4, 1701–1714.
AJ Worsey and Gerald Farin, An
-dimensional Clough-Tocher interpolant, Constructive Approximation 3 (1987), no. 1, 99–110.