#### Abstract

A local Lagrange interpolation scheme using bivariate splines of degree seven over a checkerboard triangulated quadrangulation is constructed. The method provides optimal order approximation of smooth functions.

#### 1. Introduction

Suppose that is a regular triangulation of a connected polygonal domain . For integers , we define where is the dimensional space of bivariate polynomials of degree . For integers , we further define the following super spline space: where, as usual, means that all polynomials on triangles sharing the vertex have common derivatives up to order at that vertex.

We consider the following Lagrange interpolation problem.

Problem 1. Let be a set of points in the plane, and let be a quadrangulation with vertices at the points of . Find a triangulation of and a set of additional points such that, for every choice of the data , there is a unique spline satisfying
We call a Lagrange interpolation set for and and a Lagrange interpolation pair. Although constructing Lagrange interpolation pairs sounds simple at first glance, it is in fact a complex problem, especially since we want a local and stable method which has linear complexity and provides optimal order approximation. In order to construct Lagrange interpolation pairs, both and must be carefully chosen.
For , the first result on local Lagrange interpolation by splines on triangulations was given by Nürnberger and Zeilfelder , where, by subdividing about half of the triangles with a Clough-Tocher split, a local Lagrange interpolation scheme for cubic splines on triangulations whose interior vertices have degree six was developed. Nürnberger and Zeilfelder  constructed a local Lagrange interpolation set for , where is the refining triangulation of Powell-Sabin type (I). Nürnberger and Zeilfelder  used a coloring algorithm to divide all the triangles in into two kinds: white triangles and black triangles, got a new triangulation through refining all the white triangles by the Clough-Tocher refinement, and then gave a local Lagrange interpolation set for . As to more results, the reader is referred to several survey papers .
For , Nürnberger et al.  constructed a local Lagrange interpolation set for . Liu and Fan  constructed a local Lagrange interpolation set for , where is the triangulation by refining some of triangles in with the double Clough-Tocher splits.
For general cases , a local Lagrange interpolation of has been proposed by Nürnberger et al. in , where , , and are taken as the following related values with .
Let be a quadrangulation of which consists of nondegenerate convex quadrilaterals. By adding one or two diagonals of each quadrilateral, some triangulated quadrangulations can be obtained. Nürnberger et al.  constructed a Lagrange interpolation scheme based on cubic splines on certain triangulations obtained from checkerboard quadrangulations. In , they also constructed a local Lagrange interpolation method based on cubic splines on certain triangulations obtained from a separable quadrangulation. Further they  described local Lagrange interpolation methods based on cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Their construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.
However, to the authors' knowledge, the local Lagrange interpolation schemes based on splines on any triangulated quadrangulation have not been developed. In this paper, we shall construct a Lagrange interpolation scheme on over the triangulation obtained by adding the two diagonals of each quadrilateral from checkerboard quadrangulations.
The paper is organized as follows. In Section 2 we introduce some notation and describe the Bernstein-Bézier representation of splines. In Section 3 we introduce the checkerboard triangulations. In Section 4 several lemmas of Lagrange minimal determining sets are established. In Section 5 the main results of construction of the Lagrange interpolation pair and error bounds for the interpolating splines are presented.

#### 2. Preliminaries

Throughout the paper we shall make extensive use of the well-known Bernstein-Bézier representation of splines. Let in with vertices , and the corresponding polynomial piece is written in the form: where are the Bernstein basis polynomials of degree associated with . As usual, we identify the Bernstein-Bézier coefficients with the set of domain points . We write for the union of the sets of domain points associated with the triangles of .

Given an integer , let We have similar definitions at the other vertices of . If is a vertex of a triangulation , we, respectively, define that the ring and the disk of radius around are the set where the union is taken over all triangles attached to .

We recall  that supposes that is a subspace of , and then is said to be a determining set for that provides that, for any for all implies that . Note that is a linear functional. The set is called a minimal determining set (MDS) for if there is no smaller determining set. Further, following , a basis for a spline space is called a stable local basis provided that constants , and exist depending only on the smallest angle in such that(1)for each , there is a vertex of for which ,(2)for all choices of the coefficient vector , Here is defined to be the set of all triangles surrounding vertex , and is defined to be the union of the , where are vertices of .

It is well known that a spline in is uniquely determined by its Bernstein-Bézier coefficient set . In order to describe smoothness conditions for splines, we recall some notations introduced in . Suppose that and are two adjoining triangles from , which share the oriented edge , and let where and are the Bernstein polynomials of degree on the triangles and , respectively. Given integers , let be linear functional defined on by These are called smoothness functionals of order . According to , a spline belongs to for some if and only if

#### 3. Checkerboard Triangulations

Definition 1 (see ). Suppose that is a quadrangulation consisting of quadrilaterals with largest interior angle less than . Suppose that the quadrilaterals can be colored black and white in such a way that any two quadrilaterals sharing an edge have the opposite color. Then we call a checkerboard quadrangulation. The triangulation which is obtained by drawing in both diagonals of all quadrilaterals will be called a checkerboard triangulation.
Let and denote the sets of black and white quadrilaterals of , respectively. Following , throughout this paper, we also assume that all interior vertices of are of degree four. This assumption can ensure that there exists such that for every interior vertex of , there is a unique quadrilateral sharing the vertex . For , let be the set of white quadrilaterals which share edges with black quadrilaterals. Let and for , and let be the total number of vertices of . As shown in Figure 1, a typical checkerboard triangulation is displayed in which the quadrilaterals in the set are shaded grey. It is noted that the other black quadrilaterals have not been colored.

#### 4. Lagrange Minimal Determining Sets

Lemma 2 (see ). The set of all domain points in a triangle is a Lagrange minimal determining set for the space .

Let be a quadrilateral with vertices in counterclockwise order and be the triangulated quadrangulation of with being the intersection point of the two diagonals of . As shown in Figure 2, let and for , where .

Lemma 3. The set is a stable Lagrange MDS for . These domain points are marked with in Figure 2.

Proof. Equivalently, we consider the related homogenous interpolation problem. By Lemma 2, all Bernstein-Bézier coefficients of associated to 36 domain points in must be zero. By the smoothness conditions on the edge and the smoothness conditions at vertex , all Bernstein-Bézier coefficients of must be zero except for 14 coefficients (), , , and . Let then the homogenous Lagrange interpolation conditions at the associated 14 domain points lead to where
It is easy to see that . Thus all the remaining 14 Bernstein-Bézier coefficients of are zero. Similarly, all the Bernstein-Bézier coefficients of must be also zero by smoothness conditions and given Lagrange interpolation conditions. Further, using smoothness conditions along two edges and and smoothness conditions at vertices and , all Bernstein-Bézier coefficients of are zero except for . Using the interpolation condition at domain point : , that is; since . Therefore all B coefficients of are zero.
The construction in Lemma 3 is stable in the sense that the maximum coefficient of is bounded by , where is a constant depending only on the smallest angle in . Thus, we say that is a stable Lagrange MDS for . The proof is completed.

In the following lemmas, we will consider four cases depending on how many edges of adjoin with the other .

Lemma 4. Suppose that consists of two triangulated quadrangulations and sharing the edge , where and . Let and be the points where the two diagonals of and intersect, respectively. Let the set where is the set defined in Lemma 3 for the triangulated quadrangulations of and be a stable Lagrange MDS for . These domain points are marked with in Figure 3.

Proof. Using Lemma 3, we can see that all of the B coefficients of associated with domain points in are uniquely determined by the data. Writing in B form and using , , and smoothness conditions, all B coefficients are determined except for the B coefficients associated with . Because , where
It is easy to see that . Thus, all of the remaining B coefficients of are uniquely determined by the data. Then, using Lemma 3 again, it is easy to see that all of the B coefficients of associated with domain points in are uniquely and stable determined by the data. Therefore, is a stable Lagrange MDS for . The proof is completed.

Lemma 5. Suppose that consists of three triangulated quadrangulations , , and as in Figure 4, where and . Let the set where is the set defined in Lemma 3 for the triangulated quadrangulations of and be a stable Lagrange MDS for . These domain points are marked with in Figure 4.

Proof. Using Lemma 3, we can see that all of the B coefficients of associated with domain points in and are uniquely determined by the data. Arguing as in Lemma 4, all the B coefficients of associated with domain points are uniquely determined by the data too. Writing in B form and using , and smoothness conditions, all B coefficients are determined except for . The Lagrange interpolation and and smoothness conditions at imply that these coefficients must satisfy the linear system , where where . The determinant of the matrix is By the geometric meaning of the and , we have and . Thus . So, all B coefficients of are uniquely determined. By the same principle, all B coefficients of are uniquely determined. Using Lemma 3 again, it is easy to see that all B coefficients of are uniquely and stable determined by the data. Therefore is a stable Lagrange MDS for . The proof is completed.

Lemma 6. Suppose that consists of three triangulated quadrangulations , , and as in Figure 5, where and . Let the set where is the set defined in Lemma 3 for the triangulated quadrangulations of and be a stable Lagrange MDS for . These domain points are marked with in Figure 5.

Proof. Using Lemma 3 and , , and smoothness conditions, we can see that all of the B coefficients of associated with domain points in and are uniquely determined by the data. Writing in B form and using , , and smoothness conditions, all B coefficients are determined except for the B coefficients of associated. The Lagrange interpolation and and smoothness conditions at and imply that these coefficients must satisfy the linear system , where where and . The determinant of the matrix G is By the geometric meaning of the and , we have and . Thus . So, all B coefficients of are uniquely determined. Applying Lemmas 3 and 4, it is easy to see that all B coefficients of are uniquely and stable determined by the data. Therefore is a stable Lagrange MDS for . The proof is completed.

Lemma 7. Suppose that consists of three triangulated quadrangulations , , , and as in Figure 6, where and . Let the set where is the set defined in Lemma 3 for the triangulated quadrangulations of and be a stable Lagrange MDS for . These domain points are marked with in Figure 6.

Proof. Using Lemma 3 and , , and smoothness conditions, we can see that all of the B coefficients of associated with domain points in , , and are uniquely determined by the data. Writing in B form and using , , and smoothness conditions, all B-coefficients are determined except for the B-coefficients of associated. The Lagrange interpolation and , smoothness conditions at and imply that these coefficients must satisfy the linear system , where
where , . The determinant of the matrix G is By the geometric meaning of the , , , and , we have , , , and . Thus . So, all B coefficients of are uniquely determined. Applying Lemmas 3 and 4, it is easy to see that all B coefficients of are uniquely and stable determined by the data. Therefore, is a stable Lagrange MDS for . The proof is completed.

#### 5. Construction of a Lagrange Interpolation Pair

Theorem 8. Suppose that is checkerboard triangulation. Then (32) Moreover, the following set of domain points is a stable Lagrange MDS:(1)if , choose points as in Lemma 3,(2)if , choose points as in Lemma 3, leaving out the points in the sets whenever is a vertex of which is interior to ,(3)suppose that , and let for , where and is the point where the two diagonals of intersect:(a)if , choose points as in Lemma 4,(b)if , choose points as in Lemmas 5 and 6,(c)if , choose points as in Lemma 7,(d)if sharing four edges with black quadrilaterals, say , choose the points , , , and .

Proof. To establish that is a Lagrange MDS, suppose and that we are given values for for all . We need to show that all of the B coefficients of are uniquely determined. By Lemma 3, all B-coefficient associated with domain points lying in quadrilaterals is uniquely determined. Now consider . For each vertex which is an interior vertex of , the B coefficients corresponding to domain points in the disk are already uniquely determined by , and continuity from the neighboring pieces. Leaving the corresponding basis functions out, we can then argue exactly as in Lemma 3 to see that all B coefficients of corresponding to the remaining domain points in are uniquely determined.
Now suppose that . If , then, using Lemma 4, we can see that all B coefficients of corresponding to the remaining domain points in are uniquely determined. If , then, using Lemmas 5 and 6, we can see that all B-coefficients of corresponding to the remaining domain points in are uniquely determined. If , then using Lemma 7, we can see that all B coefficients of corresponding to the remaining domain points in are uniquely determined. If shares four edges with black quadrilaterals, we can then argue exactly as in Lemma 7 to see that all B coefficients of corresponding to the remaining domain points in are uniquely determined. Since we have shown that is a MDS, it follows that .
Finally, we note that all of the above computations are stable in the sense that the size of the computed B coefficient is bounded by a constant depending only on the smallest angle in the triangulation . This follows from that the computations of Lemmas 27 are stable, and the fact that computing coefficients from , and smoothness conditions is automatically stable.

According to the Theorem 8, we have presented a Lagrange MDS for Figure 1 (see Figure 7).

Theorem 9. Let be a checkerboard triangulation, and let be the set defined in Theorem 8. Then there exists a stable local basis of , where .

Proof. It is clear that if is a MDS for , then, for each , there exists a unique spline , satisfying , for all . The splines obviously form a basis for ( ), which is commonly called the dual basis corresponding to .
The proof of Theorem 8 shows that all B coefficients of are uniquely and stable determined. So we should give the support of . As in , suppose that lies in a quadrilateral . Then we claim that(1)supp if ,(2)supp if ,(3)supp otherwise.

We are now ready to discuss interpolation. Suppose that is a checkerboard triangulation, and that are the dual basis function of Theorem 9 corresponding to Lagrange MDS for defined in Theorem 8. Then for every , there is a unique spline which satisfies