C¹-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation

Abstract

$C^1$ continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are given in order to show the performance of the approximation scheme.

1. Introduction

A novel, non-standard technique for constructing bivariate quasi-interpolating splines over uniform partitions was proposed by T. Sorokina and F. Zeilfelder in [1,2] (see also [3,4]). The essential idea of this methodology is to define the quasi-interpolant by directly providing the coefficients of the Bernstein–Bézier (BB-) form of its restriction to each of the subsets forming the partition.

In [2], the construction of $C^1$ quartic quasi-interpolants over a type-1 triangulation is addressed, so that the largest polynomial space is reproduced, namely the space ${\mathbb{P}}_3$ of polynomials of total degree less than or equal to three (see Figure 1). The coefficients of the quasi-interpolant on each triangle are linear combinations of function values at vertices and midpoints in a neighborhood of the triangle. The quasi-interpolant is constructed from them.

In [1], the same strategy is applied to construct $C^1$ quadratic quasi-interpolants on a triangulation which the authors called of type-2. Starting from a decomposition of the plane into squares, each of them is divided into eight micro-triangles by means of its diagonals and the straight lines parallel to the coordinate axes passing through the center of the square (see Figure 1).

The problem addressed in [2] is studied in detail in [5], proving that the approximation scheme proposed in [2] is a particular choice in a 19-parametric family of schemes. Moreover, different strategies for assigning values to the parameters are provided. In both [2] and [5], the quasi-interpolating splines interpolate the values at the vertices and the masks associated with the domain points that are key to the construction are applied taking into account the symmetries of the triangulation involved. Since the triangulation is uniform, these masks are independent of the specific triangle on which the quasi-interpolant is calculated (see also [6]).

Later, the cubic case was dealt with in [7], on the same triangulation used to construct quartic quasi-interpolants. The aim was to construct a $C^1$ cubic one, exact on ${\mathbb{P}}_2$, from the values at vertices and midpoints. Since it is not possible to define a quasi-interpolant that interpolates values at vertices, the authors opted to find specific masks for key domain points, including vertices, without imposing any symmetry. It was proved that there are unique masks that satisfy the required properties. Not being possible to achieve exactness on ${\mathbb{P}}_3$, this paper presents a construction on a refinement of the initial type-1 triangulation in order to achieve the optimal approximation order. Specifically, we work on a Clough–Tocher (CT-) refinement [8], which produces a subdivision into six micro-triangles of each square formed by two macro-triangles sharing an edge.

The rest of the paper is structured as follows. In Section 2, a type-1 triangulation endowed with a Clough–Tocher refinement is introduced, as well as the space of $C^1$ cubic splines defined over it. Further, a partition of the domain points associated with the micro-triangles is provided. In Section 3, the construction of quasi-interpolating splines is given and the general solution of the resulting problem. In Section 4, a method for selecting parameters based on the minimization of an upper bound of the quasi-interpolation error associated with the quartic monomials is proposed. In Section 5, the results of some numerical tests are given to illustrate the performance of the quasi-interpolation operator relative to the selected parameters. Finally, some details are included in Appendix A.

2. Bernstein–Bézier Form of Cubic Splines on a Type-1 Triangulation

Let us suppose that the triangulation is spanned by the vectors $e_1:=\left(h,h\right)$ and $e_2:=\left(h,-h\right)$, with $h>0$. Its vertices are $v_{i,j}:=i{e}_1+j{e}_2$, which define the lattice $\mathcal{V}:=\left\{v_{i,j},i,j\in \mathbb{Z}\right\}$. These vertices define squares which can be decomposed into the triangles ${\mathbb{T}}_{i,j}〈v_{i,j},v_{i+1,j+1},v_{i+1,j}〉$ and ${\mathbb{B}}_{i,j}〈v_{i,j},v_{i+1,j+1},v_{i,j+1}〉$ (see Figure 2). Therefore, a type-1 triangulation results:

$$\mathsf{\Delta }:=\bigcup _{i,j\in \mathbb{Z}}\left({\mathbb{T}}_{i,j}\cup {\mathbb{B}}_{i,j}\right).$$

When there is no need to distinguish between the types of triangles in $\mathsf{\Delta }$, we denote by $\mathbb{T}$ any one of them.

To define the refinement of $\mathsf{\Delta }$ to be used, let

$$t_{i,j}:=\frac{1}{3}\left(v_{i,j}+v_{i+1,j+1}+v_{i+1,j}\right)\mathrm{and}{b}_{i,j}:=\frac{1}{3}\left(v_{i,j}+v_{i+1,j+1}+v_{i,j+1}\right)$$

be the barycenters of ${\mathbb{T}}_{i,j}$ and ${\mathbb{B}}_{i,j}$, respectively. Then, the CT-refinement of each triangle is obtained by joining its vertices with its barycenter [8]. Each macro-triangle ${\mathbb{T}}_{i,j}$ and ${\mathbb{B}}_{i,j}$ is, respectively, divided into the following micro-triangles:

$$\begin{array}{ccc}{t}_1^{+}=〈v_{i,j},v_{i+1,j+1},t_{i,j}〉,\hfill & t_2^{+}=〈v_{i+1,j+1},v_{i+1,j},t_{i,j}〉,\hfill & t_3^{+}=〈v_{i+1,j},v_{i,j},t_{i,j}〉,\hfill \\ t_1^{-}=〈v_{i,j},v_{i,j+1},b_{i,j}〉,\hfill & t_2^{-}=〈v_{i,j+1},v_{i+1,j+1},b_{i,j}〉,\hfill & t_3^{-}=〈v_{i+1,j+1},v_{i,j},b_{i,j}〉.\hfill \end{array}$$

(1)

They are shown in Figure 2, bottom, where any reference to the subscripts of the micro-triangles has been avoided. As in the case of macro-triangles, the lower case letter t will be used to represent any of the micro-triangles of ${\mathsf{\Delta }}_{\mathrm{CT}}$.

In this paper, we consider the space of $C^1$ cubic splines on ${\mathsf{\Delta }}_{\mathrm{CT}}$ defined by

$$S_3^1\left({\mathsf{\Delta }}_{\mathrm{CT}}\right):=\left\{s\in C^1\left({\mathbb{R}}^2\right):s_{\mid t}\in {\mathbb{P}}_3\mathrm{for}\mathrm{all}t\in {\mathsf{\Delta }}_{\mathrm{CT}}\right\}.$$

where the restriction is $s_{\mid t}$ of $s\in S_3^1\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$ to a micro-triangle $t=〈V_1,V_2,V_3〉\in {\mathsf{\Delta }}_{\mathrm{CT}}$ a cubic polynomial, it can be represented using the cubic Bernstein polynomials

$$B_{\beta ,t}\left(p\right):=\frac{3!}{\beta !}{\tau }^{\beta }=\frac{6}{{\beta }_1!{\beta }_2!{\beta }_3!}{\tau }_1^{{\beta }_1}{\tau }_2^{{\beta }_2}{\tau }_3^{{\beta }_3},$$

where the multi-index notations $\beta :=\left({\beta }_1,{\beta }_2,{\beta }_3\right)\in {\mathbb{N}}_0^3$, $\left|\beta \right|:={\beta }_1+{\beta }_2+{\beta }_3$ and $\beta !:={\beta }_1!{\beta }_2!{\beta }_3!$ have been used, and $\tau :=\left({\tau }_1,{\tau }_2,{\tau }_3\right)$ provides the barycentric coordinates of point $p\in {\mathbb{R}}^2$ with respect to t, i.e., $p={\sum }_{i=1}^3{\tau }_i{V}_i$ and ${\sum }_{i=1}^3{\tau }_i=1$. The coordinates ${\tau }_1$, ${\tau }_2$ and ${\tau }_3$ are non-negative whenever p belongs to t.

Every polynomial $q\in {\mathbb{P}}_3$ can be expressed on t in terms of the cubic Bernstein basis polynomials $B_{\beta ,t}$, $\left|\beta \right|=3$, i.e., there exist values $b_{\beta }$ such that

$$q\left(x,y\right)=q\left(\tau \right)=\sum _{\left|\beta \right|=3}{b}_{\beta ,t}{B}_{\beta ,t}\left(\tau \right).$$

Coefficients in $D_t:=\left\{b_{\beta ,t},\left|\beta \right|=3\right\}$ are said to be the Bernstein–Bézier (BB-) coefficients of q. They are linked to the domain points ${\xi }_{\beta ,t}$ determined by the barycentric coordinates $\left(\frac{{\beta }_{1,t}}{3},\frac{{\beta }_{2,t}}{3},\frac{{\beta }_{3,t}}{3}\right)$ with respect to t. They determine the lattice $L_3\left(t\right)$. The graph of q on t is included in the convex hull of $\left\{\left({\xi }_{\beta ,t},b_{\beta ,t}\right),\left|\beta \right|=3\right\}$.

On each micro-triangle, an element $s\in S_3^1\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$ is uniquely determined by ten BB-coefficients, associated with the corresponding domain points. When all macro-triangles are taken into account, a subset of domain points is obtained, which we note ${\mathcal{D}}_3\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$, i.e., ${\mathcal{D}}_3\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)={\bigcup }_{t\in {\mathsf{\Delta }}_{\mathrm{CT}}}{L}_3\left(t\right)$, where the union is formed without taking repetitions into account. To determine s, it is necessary to give the BB-coefficients associated with all the points of ${\mathcal{D}}_3\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$. As the triangulation is uniform, following the approach in [2,4,5,6], it is sufficient to establish a partition $\left\{D_{i,j},i,j\in \mathbb{Z}\right\}$ of ${\mathcal{D}}_3\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$ and define the BB-coefficients linked to the domain points in $D_{i,j}$.

Figure 3 shows the twenty-seven domain points forming $D_{i,j}$, which are linked to vertex $v_{i,j}$. Each of them has the subscripts of $v_{i,j}$. The vertices and barycenter have already been defined. They remaining domain points in $D_{i,j}$ are given next:

$$\begin{array}{cccc}\hfill u_{i,j}^{1,1}:=& \frac{1}{3}(2v_{i,j}+v_{i+1,j+1}),\hfill & \hfill u_{i,j}^{1,0}:=& \frac{1}{3}(2v_{i,j}+v_{i+1,j}),\hfill \\ \hfill u_{i,j}^{2,1}:=& \frac{1}{3}(2v_{i,j}+t_{i,j}),\hfill & \hfill u_{i,j}^{-1,-1}:=& \frac{1}{3}(2v_{i,j}+v_{i-1,j-1}),\hfill \\ \hfill u_{i,j}^{1,-1}:=& \frac{1}{3}(2v_{i,j}+b_{i,j-1}),\hfill & \hfill u_{i,j}^{0,-1}:=& \frac{1}{3}(2v_{i,j}+v_{i,j-1}),\hfill \\ \hfill u_{i,j}^{-1,-2}:=& \frac{1}{3}(2v_{i,j}+t_{i-1,j-1}),\hfill & \hfill u_{i,j}^{-2,-1}:=& \frac{1}{3}(2v_{i,j}+b_{i-1,j-1}),\hfill \\ \hfill u_{i,j}^{-1,0}:=& \frac{1}{3}(2v_{i,j}+v_{i-1,j}),\hfill & \hfill u_{i,j}^{-1,1}:=& \frac{1}{3}(2v_{i,j}+t_{i-1,j}),\hfill \\ \hfill u_{i,j}^{0,1}:=& \frac{1}{3}(2v_{i,j}+v_{i,j+1}),\hfill & \hfill u_{i,j}^{1,2}:=& \frac{1}{3}(2v_{i,j}+b_{i,j}),\hfill \\ \hfill x_{i,j}^{1,1}:=& \frac{1}{3}(v_{i,j}+v_{i+1,j+1}+b_{i,j}),\hfill & \hfill x_{i,j}^{1,0}:=& \frac{1}{3}(v_{i,j}+v_{i+1,j}+b_{i,j-1}),\hfill \\ \hfill x_{i,j}^{0,1}:=& \frac{1}{3}(v_{i,j}+v_{i,j+1}+b_{i,j}),\hfill & \hfill y_{i,j}^{2,1}:=& \frac{1}{3}(v_{i,j}+2t_{i,j}),\hfill \\ \hfill y_{i,j}^{1,-1}:=& \frac{1}{3}(v_{i,j}+2b_{i,j-1}),\hfill & \hfill y_{i,j}^{-1,-2}:=& \frac{1}{3}(v_{i,j}+2t_{i-1,j-1}),\hfill \\ \hfill y_{i,j}^{-2,-1}:=& \frac{1}{3}(v_{i,j}+2b_{i-1,j-1}),\hfill & \hfill y_{i,j}^{-1,1}:=& \frac{1}{3}(v_{i,j}+2t_{i-1,j})\hfill \\ \hfill y_{i,j}^{1,2}:=& \frac{1}{3}(v_{i,j}+2b_{i,j}),\hfill & \hfill z_{i,j}^{1,1}:=& \frac{1}{3}(v_{i,j}+v_{i+1,j+1}+t_{i,j}),\hfill \\ \hfill z_{i,j}^{1,0}:=& \frac{1}{3}(v_{i,j}+v_{i+1,j}+t_{i,j}),\hfill & \hfill z_{i,j}^{0,1}:=& \frac{1}{3}(v_{i,j}+v_{i,j+1}+t_{i-1,j}).\hfill \end{array}$$

Figure 4 shows the domain points in D lying in the hexagon formed by the six triangles sharing the vertex $v_{i,j}$.

3. ${\mathit{C}}^{\mathbf{1}}$ Quasi-Interpolating Splines on a Clough–Tocher Refinement

The main objective of this work is to construct a quasi-interpolation operator for $S_3^1\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$ that is exact on ${\mathbb{P}}_3$ in order to improve the result obtained in [7]. Let us denote it as $\mathcal{Q}$. It is assumed that the values of a function f are known at the domain points in ${\mathcal{D}}_3\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$.

The quasi-interpolant $\mathcal{Q}f\in S_3^1\left({\mathsf{\Delta }}_{\mathrm{CT}}\right)$ of f should be constructed in such a way that the BB-coefficients of the restriction $\mathcal{Q}{f}_{\mid t}$ to each micro-triangle $t\in {\mathsf{\Delta }}_{\mathrm{CT}}$ are defined as combinations of those values of f. In other words, $\mathcal{Q}{f}_{\mid t}$ is written in the basis of Bernstein polynomials $B_{\beta ,t}$, $\left|\beta \right|=3$, as

$$\mathcal{Q}{f}_{\mid t}=\sum _{\gamma \in {\mathsf{\Delta }}_3}{P}_{\gamma }{B}_{\gamma ,t},$$

where $P_{\gamma }$ denotes the BB-coefficient associated with the domain point $p_{\gamma }\in t$, ${\mathsf{\Delta }}_3$ is the set of indices with length equal to 3 written in the lexicographical order, i.e.,

$$\begin{array}{cc}\hfill {\mathsf{\Delta }}_3& =\left\{\left(3,0,0\right),\left(2,1,0\right),\left(2,0,1\right),\left(1,2,0\right),\left(1,1,1\right),\right.\hfill \\ \hfill & \left.\left(1,0,2\right),\left(0,3,0\right),\left(0,2,1\right),\left(0,1,2\right),\left(0,0,3\right)\right\}\hfill \end{array}$$

and the vertices of each micro-triangle follow in the order they appear in (1).

For instance, with regard to the micro-triangle $t_1^{+}$ of ${\mathbb{T}}_{i,j}$ (see Figure 5) we write

$$\begin{array}{cc}\hfill \mathcal{Q}{f}_{\mid t_1^{+}}& =V_{i,j}{B}_{(3,0,0),t_1^{+}}+U_{i,j}^{1,1}{B}_{(2,1,0),t_1^{+}}+U_{i,j}^{2,1}{B}_{(2,0,1),t_1^{+}}+U_{i+1,j+1}^{-1,-1}{B}_{(1,2,0),t_1^{+}}\hfill \\ \hfill & +Z_{i,j}^{1,1}{B}_{(1,1,1),t_1^{+}}+Y_{i,j}^{2,1}{B}_{(1,0,2),t_1^{+}}+V_{i+1,j+1}{B}_{(0,3,0),t_1^{+}}\hfill \\ \hfill & +U_{i+1,j+1}^{-1,-2}{B}_{(0,2,1),t_1^{+}}+Y_{i+1,j+1}^{-1,-2}{B}_{(0,1,2),t_1^{+}}+T_{i,j}{B}_{(0,0,3),t_1^{+}}.\hfill \end{array}$$

Similar expressions are obtained for the restrictions of $\mathcal{Q}f$ to the other two micro-triangles of ${\mathbb{T}}_{i,j}$ and those three into which ${\mathbb{B}}_{i,j}$ is divided.

The BB-coefficients involved in the definition of $\mathcal{Q}f$ on each micro-triangle of ${\mathbb{T}}_{i.j}$ and ${\mathbb{B}}_{i,j}$ will be linear combinations of f at the specific domain points for cubic polynomials lying in the hexagon defined by the triangles sharing vertex $v_{i,j}$. Specifically, the union without repetitions ${\mathcal{D}}_3\left(\mathsf{\Delta }\right):={\bigcup }_{\mathbb{T}\in \mathsf{\Delta }}{L}_3\left(\mathbb{T}\right)$ is formed and decomposed as

$${\mathcal{D}}_3\left(\mathsf{\Delta }\right)=\bigcup _{i,j\in \mathbb{Z}}{S}_{i,j},$$

where the ordered subset $S_{i,j}$ consists of the thirty-seven domain points given below:

$$\begin{array}{cc}\hfill S_{i,j}& :=\left\{v_{i,j},u_{i,j}^{1,1},u_{i,j}^{1,0},u_{i,j}^{0,-1},u_{i,j}^{-1,-1},u_{i,j}^{-1,0},u_{i,j}^{0,1},u_{i+1,j+1}^{-1,-1},t_{i,j},u_{i+1,j}^{-1,0},b_{i,j-1},\right.\hfill \\ \hfill & u_{i,j-1}^{0,-1},t_{i-1,j-1},u_{i-,j-1}^{-1,-1},b_{i-1,j-1},u_{i-1,j}^{1,0},t_{i-1,j},u_{i,j+1}^{0,-1},b_{i,j},v_{i+1,j+1},u_{i+1,j+1}^{0,-1},\hfill \\ \hfill & u_{i+1,j}^{0,1},v_{i+1,j},u_{i+1,j}^{-1,-1},u_{i,j-1}^{1,1},v_{i,j-1},u_{i,j-1}^{-1,0},u_{i-1,j-1}^{1,0},v_{i-1,j-1},u_{i-1,j-1}^{0,1},u_{i-1,j}^{0,-1},\hfill \\ \hfill & \left.v_{i-1,j},u_{i-1,j}^{1,1},u_{i,j+1}^{-1,-1},v_{i,j+1},u_{i,j+1}^{1,0},u_{i+1,j+1}^{-1,0}\right\}.\hfill \end{array}$$

The BB-coefficient P of a domain point p is a linear combination of values of f at points in $S_{i,j}$, its coefficients give rise to a vector $M\left(p\right)$, ordered as $S_{i,j}$, which is said to be the mask of p. If $f\left(S_{i,j}\right):=\left\{f\left(p\right),p\in S_{i,j}\right\}$ is also ordered as $S_{i,j}$, then

$$P=M\left(p\right)·f\left(S_{i,j}\right):=\sum _{\ell =1}^{37}M{\left(p\right)}_{\ell }f{\left(S_{i,j}\right)}_{\ell },$$

where $M{\left(p\right)}_{\ell }$ and $f{\left(S_{i,j}\right)}_{\ell }$ stand for the ℓ-th entries of $M\left(p\right)$ and $f\left(S_{i,j}\right)$, respectively.

In the following, we state the problem that is the object of this work.

Problem 1.

Find masks for the domain points in $D_{i,j}$ such that the associated quasi-interpolation operator $\mathcal{Q}$ is exact on ${\mathbb{P}}_3$ and produces $C^1$ quasi-interpolating splines.

The following result holds.

Proposition 2.

Problem 1 has a 17-parametric family of solutions.

Proof. 

Given an arbitrary function f, $C^1$ continuity of $\mathcal{Q}f$ across segment $\left[v_{i,j},v_{i+1,j}\right]$ is equivalent to the following conditions [9] (Thm. 2.28) (see Figure 6 and the notations used for the domain points in Figure 3 and Figure 4):

$$\begin{array}{cc}\hfill V_{i,j}+U_{i,j}^{1,0}-U_{i,j}^{1,-1}-U_{i,j}^{2,1}& =0,\hfill \\ \hfill U_{i,j}^{1,0}+U_{i+1,j}^{-1,0}-X_{i,j}^{1,0}-Z_{i,j}^{1,0}& =0,\hfill \\ \hfill U_{i+1,j}^{-1,0}+V_{i+1,j}-U_{i+1,j}^{2,1}-U_{i+1,j}^{-1,1}& =0.\hfill \end{array}$$

For $\left[v_{i,j},v_{i+1,j+1}\right]$,

$$\begin{array}{cc}\hfill V_{i,j}+U_{i,j}^{1,1}-U_{i,j}^{2,1}-U_{i,j}^{1,2}& =0,\hfill \\ \hfill U_{i,j}^{1,1}+U_{i+1,j+1}^{-1,-1}-X_{i,j}^{1,1}-Z_{i,j}^{1,1}& =0,\hfill \\ \hfill U_{i+1,j+1}^{-1,-1}+V_{i+1,j+1}-U_{i+1,j+1}^{-1,-2}-U_{i+1,j+1}^{2,1}& =0.\hfill \end{array}$$

And for $\left[v_{i,j},v_{i,j+1}\right]$,

$$\begin{array}{cc}\hfill V_{i,j}+U_{i,j}^{0,1}-U_{i,j}^{-1,1}-U_{i,j}^{1,2}& =0,\hfill \\ \hfill U_{i,j}^{0,1}+U_{i,j+1}^{0,-1}-Z_{i,j}^{0,1}-X_{i,j}^{0,1}& =0,\hfill \\ \hfill U_{i,j+1}^{0,-1}+V_{i,j+1}-U_{i,j+1}^{-1,-2}-U_{i,j+1}^{1,-1}& =0.\hfill \end{array}$$

Regarding micro-edges, $C^1$ continuity across $\left[v_{i,j},t_{i,j}\right]$ is equivalent to conditions

$$U_{i,j}^{2,1}-{\displaystyle \frac{1}{3}}\left(V_{i,j}+U_{i,j}^{1,0}+U_{i,j}^{1,1}\right)=0,Y_{i,j}^{2,1}-{\displaystyle \frac{1}{3}}\left(U_{i,j}^{2,1}+Z_{i,j}^{1,0}+Z_{i,j}^{1,1}\right)=0.$$

Similarly, it is satisfied across $\left[v_{i+1,j},t_{i,j}\right]$ and $\left[v_{i+1,j+1},t_{i,j}\right]$, respectively, if and only if

$$\begin{array}{cc}\hfill U_{i+1,j}^{-1,1}-{\displaystyle \frac{1}{3}}\left(V_{i+1,j}+U_{i+1,j}^{-1,0}+U_{i+1,j}^{0,1}\right)& =0,\hfill \\ \hfill Y_{i+1,j+1}^{-1,-1}-{\displaystyle \frac{1}{3}}\left(U_{i+1,j}^{-1,1}+Z_{i,j}^{1,0}+Z_{i+1,j}^{0,1}\right)& =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill U_{i+1,j+1}^{-1,-2}-{\displaystyle \frac{1}{3}}\left(V_{i+1,j+1}+U_{i+1,j+1}^{-1,-1}+U_{i+1,j+1}^{0,-1}\right)& =0,\hfill \\ \hfill Y_{i+1,j+1}^{-1,-2}-{\displaystyle \frac{1}{3}}\left(U_{i+1,j+1}^{-1,-2}+Z_{i,j}^{1,1}+Z_{i+1,j}^{0,1}\right)& =0.\hfill \end{array}$$

For the micro-sides of macro-triangle ${\mathbb{B}}_{i,j}$, six new conditions are involved. For $\left[v_{i,j},b_{i,j}\right]$, $\left[v_{i,j+1},b_{i,j}\right]$ and $\left[v_{i+1,j+1},b_{i,j}\right]$, $C^1$ regularity is equivalent to

$$U_{i,j}^{1,2}-\frac{1}{3}\left(V_{i,j}+U_{i,j}^{0,1}+U_{i,j}^{1,1}\right)=0,Y_{i,j}^{1,2}-\frac{1}{3}\left(U_{i,j}^{1,2}+X_{i,j}^{1,1}+X_{i,j}^{0,1}\right)=0,$$

$$\begin{array}{cc}\hfill U_{i,j+1}^{1,-1}-\frac{1}{3}\left(V_{i,j+1}+U_{i,j+1}^{0,-1}+U_{i,j+1}^{1,0}\right)& =0,\hfill \\ \hfill Y_{i,j+1}^{1,-1}-\frac{1}{3}\left(U_{i,j+1}^{1,-1}+X_{i,j}^{0,1}+X_{i,j+1}^{1,0}\right)& =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill U_{i+1,j+1}^{2,-1}-\frac{1}{3}\left(V_{i+1,j+1}+U_{i+1,j+1}^{-1,-1}+U_{i+1,j+1}^{-1,0}\right)& =0,\hfill \\ \hfill Y_{i+1,j+1}^{-2,-1}-\frac{1}{3}\left(U_{i+1,j+1}^{2,1}+X_{i,j}^{1,1}+X_{i,j+1}^{1,0}\right)& =0,\hfill \end{array}$$

respectively. Finally, $C^1$ continuity at the barycenters of ${\mathbb{T}}_{i,j}$ and ${\mathbb{B}}_{i,j}$ is obtained if and only if

$$\begin{array}{cc}\hfill T_{i,j}-\frac{1}{3}\left(Y_{i,j}^{2,1}+Y_{i+1,j+1}^{-1,-2}+Y_{i+1,j-1}^{-1,1}\right)& =0,\hfill \\ \hfill B_{i,j}-\frac{1}{3}\left(Y_{i,j}^{1,2}+Y_{i+1,j+1}^{-2,-1}+Y_{i,j+1}^{-1,-1}\right)& =0.\hfill \end{array}$$

These are all equalities involving the values $f\left(p\right)$, $p\in S_{i,j}$, so $\mathcal{Q}f$ is $C^1$ continuous if and only if all the coefficients of the f-values in these equalities are zero. Therefore, the requirements on the $C^1$ continuity are equivalent to a system of equations having a 122-parametric family of solutions. To these equations must be added those related to the exactness of the operator on ${\mathbb{P}}_3$. They are obtained by imposing that the BB-coefficients on each microtriangle of the monomials of degree less than or equal to three and those of their quasi-interpolants are equal. The resulting system can be solved with a Computer Algebra System, namely, Mathematica, obtaining the existence of a 17-parametric family of solutions. The free parameters are entries with indices 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 19, 20, 21, and 22 of the mask $M\left(b_{i,j}\right)$. □

Figure 7 shows the mask relative to vertex $v_{i,j}$. The entries of the masks for $u_{i,j}^{0,-1}$, $u_{i,j}^{-1,-2}$, $u_{i,j}^{-2,-1}$ and $u_{i,j}^{-1,0}$ are almost all zero, and the following expressions for their BB-coefficients are found:

$$\begin{array}{cc}\hfill U_{i,j}^{-1,0}& =-\frac{5}{6}f\left(v_{i,j}\right)+3f\left(u_{i,j}^{-1,0}\right)-\frac{3}{2}f\left(u_{i,j-1}^{0,-1}\right)+\frac{1}{3}f\left(v_{i,j-1}\right),\hfill \\ \hfill U_{i,j}^{0,-1}& =-\frac{5}{6}f\left(v_{i,j}\right)-\frac{3}{2}f\left(u_{i,j}^{-1,0}\right)+3f\left(u_{i,j-1}^{0,-1}\right)+\frac{1}{3}f\left(v_{i+1,j+1}\right),\hfill \\ \hfill U_{i,j}^{-1,-2}& =-\frac{5}{6}f\left(v_{i,j}\right)+f\left(u_{i,j}^{-1,0}\right)+2f\left(u_{i,j}^{0,-1}\right)-\frac{1}{2}f\left(u_{i,j-1}^{0,-1}\right)\hfill \\ \hfill & -f\left(u_{i+1,j}^{-1,0}\right)+\frac{2}{9}f\left(v_{i+1,j+1}\right)+\frac{1}{9}f\left(v_{i,j-1}\right),\hfill \\ \hfill U_{i,j}^{-2,-1}& =-\frac{5}{6}f\left(v_{i,j}\right)+2f\left(u_{i,j}^{-1,0}\right)+f\left(u_{i,j}^{0,-1}\right)-f\left(u_{i,j-1}^{0,-1}\right)\hfill \\ \hfill & -\frac{1}{2}f\left(u_{i+1,j}^{-1,0}\right)+\frac{1}{9}f\left(v_{i+1,j+1}\right)+\frac{2}{9}f\left(v_{i,j-1}\right).\hfill \end{array}$$

Fourteen of the masks relative to the remaining twenty-two domain points in $D_{i,j}$ do not depend on any parameters and appear in Appendix A. Those of $t_{i,j}$, $b_{i,j}$, $y_{i,j}^{1,2}$, $y_{i,j}^{2,1}$, $y_{i,j}^{-1,-2}$, $y_{i,j}^{-2,-1}$, $x_{i,j}^{1,1}$, and $z_{i,j}^{1,1}$ have very long entries and will not be given.

Remark 1.

It can be proved that it is not possible to obtain quasi-interpolants with the required characteristics if the BB-coefficients are linear combinations of function values at the vertices lying in the hexagon $H_{i,j}$ determined by the six triangles sharing vertex $v_{i,j}$, and the midpoints of the edges of $H_{i,j}$. Neither is it possible to construct $C^1$ cubic quasi-interpolants exact on ${\mathbb{P}}_3$ in this way if function values at $v_{i,j}$ and at the eighteen vertices closest to it are used.

Moreover, quasi-interpolation error estimates are found using a standard procedure [2].

Proposition 3.

There exists an absolute constant K such that for every $f\in C^{m+1}\left({\mathbb{R}}^2\right)$, $0\le m\le 2$,

$${∥D^{\gamma }\left(f-\mathcal{Q}f\right)∥}_{\infty ,\mathbb{T}}\le K{h}^{m+1-\left|\gamma \right|}{∥D^{m+1}f∥}_{\infty ,{\Omega }_{\mathbb{T}}},$$

(2)

for all $0\le \left|\gamma \right|\le 1$, $\gamma =({\gamma }_1,{\gamma }_2)$, with ${\mathsf{\Omega }}_{\mathbb{T}}$ denoting the union of the triangles in $\mathsf{\Delta }$ having a non-empty intersection with T.

4. Selecting Parameters

An obvious choice is to make all parameters equal to zero. However, a reasonable strategy is to minimize an upper bound of the quasi-interpolation error for monomials of smaller degree non reproduced by the quasi-interpolation operator, namely $m_{k,4-k}\left(x,y\right):=x^k{y}^{4-k}$, $k=0,1,2,3,4$. Let us suppose that the BB-coefficients of $m_{k,4-k}$ relative to each micro-triangle $t_{\ell }^{+}$, $\ell =1,2,3$, of ${\mathbb{T}}_{i,j}$ are ${\mu }_{k,\beta ,t_{\ell }^{+}}$, $\left|\beta \right|=4$, and that those of the cubic quasi-interpolant $Q{m}_{k,4-k}$ are $b_{k,\gamma ,t_{\ell }^{+}}$, $\left|\gamma \right|=4$. By degree elevation, $Q{m}_{k,4-k\mid t_{\ell }^{+}}$ can be represented as a quartic polynomial having BB-coefficients $b_{k,\beta ,t_{\ell }^{+}}$, $\left|\beta \right|=4$, which depend on parameters $z_r:=M{\left(b_{i,j}\right)}_r$, $1\le r\le 12$, and $z_r:=M{\left(b_{i,j}\right)}_{r+5}$, $13\le r\le 17$. Therefore, the BB-coefficients of the restriction of $m_{k,4-k}-Q{m}_{k,4-k}$ to $t_{\ell }^{+}$ have the form

$${\sigma }_{k,t_{\ell }^{+}}\left(z\right)=c_{k,t_{\ell }^{+}}+\sum _{r=1}^{17}{c}_{k,t_{\ell }^{+}}^{\left(r\right)}{z}_r$$

for real values $c_{k,t_{\ell }^{+}}$ and $c_{k,t_{\ell }^{+}}^{\left(r\right)}$, where $z:=\left(z_1,\dots ,z_{17}\right)$. Since the Bernstein polynomials relative to $t_{\ell }^{+}$ form a partition of unity, then the infinity norm of $m_{k,4-k}-Q{m}_{k,4-k}$ is bounded by

$$max\left\{\left|{\sigma }_{k,t_{\ell }^{+}}\left(z\right)\right|,\ell =1,2,3\right\}.$$

Consequently, an upper bound for the quasi-interpolation errors for quartic monomials in the macro-triangle ${\mathbb{T}}_{i,j}$ is

$$U_{+}\left(z\right):=max\left\{\left|{\sigma }_{k,t_{\ell }^{+}}\left(z\right)\right|,\ell =1,2,3;k=0,1,2,3,4\right\}.$$

Analogously, an upper bound of such errors in the macro-triangle ${\mathbb{B}}_{i,j}$ is written as

$$U_{-}\left(z\right):=max\left\{\left|{\sigma }_{k,t_{\ell }^{-}}\left(z\right)\right|,\ell =1,2,3;k=0,1,2,3,4\right\},$$

where

$${\sigma }_{k,t_{\ell }^{-}}\left(z\right)=c_{k,t_{\ell }^{-}}+\sum _{r=1}^{17}{c}_{k,t_{\ell }^{.}}^{\left(r\right)}{z}_r,$$

for real values $c_{k,t_{\ell }^{-}}$ and $c_{k,t_{\ell }^{.}}^{\left(r\right)}$. In short, the function

$$U\left(z\right):=max\left\{U_{+}\left(z\right),U_{-}\left(z\right)\right\}$$

is an upper bound for the quasi-interpolation errors for quartic monomials in the square ${\mathbb{T}}_{i,j}\bigcup {\mathbb{B}}_{i,j}$.

Function U can be rewritten as

$$U\left(z\right)=\underset{1\le \alpha \le 30}{max}\frac{1}{c_{\alpha }}\left(d_{\alpha }+\sum _{\beta =1}^{17}{e}_{\alpha ,\beta }\left|f_{\alpha ,\beta }·z\right|\right),$$

where $c_{\alpha },d_{\alpha },e_{\alpha ,\beta }\in \mathbb{N}$, $f_{\alpha ,\beta }\in {\mathbb{Z}}^{17}$ and $A·B:={\sum }_{s=1}^{17}{A}_s{B}_s$. The number of terms involved in each sum depends on $\alpha$, because some of them will be zero. Therefore, the minimization of U is equivalent to the following linear programming problem:

$$\begin{array}{cc}\mathrm{Minimize}& \mu \hfill \\ \mathrm{such}\mathrm{that}& \left\{\begin{array}{cc}{d}_{\alpha }+\sum _{\beta =1}^{17}{e}_{\alpha ,\beta }\left(u_{\alpha ,\beta }+v_{\alpha ,\beta }\right)-c_{\alpha }\mu \le 0,\hfill & 1\le \alpha \le 30,\hfill \\ f_{\alpha ,\beta }·\left(Z^{+}-Z^{-}\right)-u_{\alpha ,\beta }+v_{\alpha ,\beta }=0,\hfill & 1\le \alpha \le 30,1\le \beta \le 17,\hfill \\ u_{p,n},v_{p,n},X_1,X_2,Y_1,Y_2,Z_1,Z_2,\mu \ge 0,\hfill \end{array}\right.\hfill \end{array}$$

where it has been used that each variable $z_r$ can be written as $z_r=z_r^{+}-z_r^{-}$, $z_r^{+},z_r^{-}\ge 0$, therefore $Z=Z^{+}-Z^{-}$, with $Z^{+}:=\left(z_1^{+},\dots ,z_{17}^{+}\right)$ and $Z^{-}:=\left(z_1^{-},\dots ,z_{17}^{-}\right)$. The solution of this problem has been exactly determined by using Mathematica, and the minimum value $\mu =\frac{35971348390906381}{87945041427390}$ is reached at

$$\begin{array}{cccc}\hfill Z_3^{+}& =\frac{33654106472661220639}{24647711830550794440},\hfill & \hfill Z_6^{-}& =\frac{28931119278287059059781}{79874153306877553434720},\hfill \\ \hfill Z_7^{-}& =\frac{147713415264798351289}{49295423661101588880},\hfill & \hfill Z_9^{-}& =\frac{71687410464642966611}{49295423661101588880},\hfill \\ \hfill Z_{10}^{-}& =\frac{3723562194545339719095199}{1118238146296285748086080},\hfill & \hfill Z_{12}^{-}& =\frac{3459921708110971652593}{12288331277981162066880},\hfill \\ \hfill Z_{13}^{-}& =\frac{1437915323322245022121277}{1863730243827142913476800},\hfill & \hfill Z_{15}^{+}& =\frac{9334610941403380115035381}{10064143316666571732774720},\hfill \end{array}$$

being equal to zero all the remaining values. Therefore, the minimum is attained at point $z^{*}$ with components $z_r^{*}=0$ for $r\in \left\{1,2,4,5,8,11,14,16,17\right\}$, and

$$\begin{array}{cccc}\hfill z_3^{*}& =\frac{33654106472661220639}{24647711830550794440},\hfill & \hfill z_6^{*}& =-\frac{28931119278287059059781}{79874153306877553434720},\hfill \\ \hfill z_7^{*}& =-\frac{147713415264798351289}{49295423661101588880},\hfill & \hfill z_9^{*}& =-\frac{71687410464642966611}{49295423661101588880},\hfill \\ \hfill z_{10}^{*}& =-\frac{3723562194545339719095199}{1118238146296285748086080},\hfill & \hfill z_{12}^{*}& =-\frac{3459921708110971652593}{12288331277981162066880},\hfill \\ \hfill z_{13}^{*}& =-\frac{1437915323322245022121277}{1863730243827142913476800},\hfill & \hfill z_{15}^{*}& =\frac{9334610941403380115035381}{10064143316666571732774720}.\hfill \end{array}$$

5. Numerical Tests

In this section, the performance of the quasi-interpolation operator $Q^{*}$ defined by the masks provided by the solution above is tested. To perform this, we consider Franke’s function

$$\begin{array}{cc}\hfill f_1\left(x_1,x_2\right)& =\frac{3}{4}exp\left(-\frac{{\left(9x_1-2\right)}^2}{4}-\frac{{\left(9x_2-2\right)}^2}{4}\right)+\frac{3}{4}exp\left(-\frac{{\left(9x_1+1\right)}^2}{49}-\frac{9x_2+1}{10}\right)\hfill \\ \hfill & +\frac{1}{2}exp\left(-\frac{{\left(9x_1-7\right)}^2}{4}-\frac{{\left(9x_2-3\right)}^2}{4}\right)-\frac{1}{5}exp\left(-{\left(9x_1-4\right)}^2-{\left(9x_2-7\right)}^2\right)\hfill \end{array}$$

and Nielson’s function

$$f_2\left(x_1,x_2\right)=\frac{x_2}{2}{cos}^4\left(4(x_1^2+x_2-1)\right)$$

to produce quasi-interpolants on the unit square [10,11]. The plots of $f_1$ and $f_2$ are shown in Figure 8, together with those of their quasi-interpolants obtained by diving the unit interval into 256 equal parts.

The quasi-interpolation error is estimated as

$$\underset{k,\ell =1,\dots ,400}{max}\left|{\mathbb{Q}}^{*}f\left(x_k,y_{\ell }\right)-f\left(x_k,y_{\ell }\right)\right|,$$

$x_k$ and $y_{\ell }$ being equally spaced points in $\left[0,1\right]$. The numerical convergence order (NCO) is given by the rate

$$\mathrm{NCO}:=log\left(\frac{\mathtt{E}\left(h_2\right)}{\mathtt{E}\left(h_1\right)}\right)/log\left(\frac{h_2}{h_1}\right),$$

where $\mathtt{E}\left(h\right)$ stands for the estimated error associated with the step length h.

The quasi-interpolation errors are estimated for different values of the step length h and the NCO are calculated. The results are shown in

Table 1. Errors and NCOs for functions $f_1$ and $f_2$ with $h=1/n$, $n=20,40,80,160$.

	${\mathit{f}}_1$		${\mathit{f}}_2$
n	Estimated Error	NCO	Estimated Error	NCO
16	$7.07377\times {10}^{-1}$	–	$1.47146\times {10}^{-1}$	–
32	$4.49051\times {10}^{-2}$	$3.97753$	$1.44799\times {10}^{-2}$	$3.34512$
64	$3.14830\times {10}^{-3}$	$3.83423$	$8.62813\times {10}^{-4}$	$4.06886$
128	$1.76965\times {10}^{-4}$	$4.15304$	$5.36388\times {10}^{-5}$	$4.00770$
256	$1.07615\times {10}^{-5}$	$4.03951$	$3.48823\times {10}^{-6}$	$3.94271$

. They confirm the theoretical ones.

6. Conclusions

In this work, $C^1$ cubic quasi-interpolants have been defined on a Clough–Tocher refinement of a type-1 triangulation, providing directly their BB-coefficients on each of the micro-triangles of the sub-triangulation, which are linear combinations of the values taken by the approximated function at specific points in a neighborhood of each macro-triangle. Cubic polynomials are reproduced. The general problem has a 17-parametric family of solutions and a specific solution has been chosen, which minimizes an upper bound of the quasi-interpolation errors associated with the quartic monomials.

The results improve on those available for cubic quasi-interpolation over a type-1 triangulation since the quasi-interpolation operator is now exact on ${\mathbb{P}}_3$ instead of ${\mathbb{P}}_2$.