Local Lagrange Interpolations Using Bivariate 𝐶² Splines of Degree Seven on Triangulated Quadrangulations

Abstract

A local Lagrange interpolation scheme using bivariate 𝐶² splines of degree seven over a checkerboard triangulated quadrangulation is constructed. The method provides optimal order approximation of smooth functions.

1. Introduction

Suppose that Δ is a regular triangulation of a connected polygonal domain Ω. For integers 0≤𝑟≤𝑑, we define 𝑆^𝑟_𝑑(Δ)∶=𝑠∈𝐶^𝑟(Ω)∶𝑠∣_𝑇∈𝑃_𝑑,∀𝑇∈Δ, (1) where 𝑃_𝑑 is the ((𝑑+2)(𝑑+1))/2 dimensional space of bivariate polynomials of degree 𝑑. For integers 𝑟≤𝜌≤𝑑, we further define the following super spline space: 𝑆_𝑑^𝑟,𝜌(Δ)∶=𝑠∈𝐶^𝑟(Ω)∶𝑠∈𝐶^𝜌(𝑣),∀𝑣∈Δ, (2) where, as usual, 𝑠∈𝐶^𝜌(𝑣) means that all polynomials on triangles sharing the vertex 𝑣 have common derivatives up to order 𝜌 at that vertex.

We consider the following Lagrange interpolation problem.

Problem 1. Let 𝒱∶={𝜂_𝑖}^𝑛_𝑖=1 be a set of points in the plane, and let ◊ be a quadrangulation with vertices at the points of 𝑉. Find a triangulation Δ of ◊ and a set of additional points {𝜂_𝑖}^𝑁_𝑖=𝑛+1 such that, for every choice of the data {𝑧_𝑖}^𝑁_𝑖=1, there is a unique spline 𝑠∈𝒮 satisfying 𝑠𝜂_𝑖=𝑧_𝑖,𝑖=1,…,𝑁. (3)
We call 𝒫∶={𝜂_𝑖}^𝑁_𝑖=1 a Lagrange interpolation set for 𝒮 and 𝒫 and 𝒮 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 𝒮 and 𝒫 must be carefully chosen.
For 𝑟=1, the first result on local Lagrange interpolation by splines on triangulations was given by Nürnberger and Zeilfelder [1], where, by subdividing about half of the triangles with a Clough-Tocher split, a local Lagrange interpolation scheme for cubic 𝐶¹ splines on triangulations whose interior vertices have degree six was developed. Nürnberger and Zeilfelder [2] constructed a local Lagrange interpolation set for 𝑆¹₂(▵_PS1), where ▵_PS1 is the refining triangulation of Powell-Sabin type (I). Nürnberger and Zeilfelder [3] used a coloring algorithm to divide all the triangles in ▵ into two kinds: white triangles and black triangles, got a new triangulation ▵_CT through refining all the white triangles by the Clough-Tocher refinement, and then gave a local Lagrange interpolation set for 𝑆¹₃(▵_CT). As to more results, the reader is referred to several survey papers [4–6].
For 𝑟=2, Nürnberger et al. [7] constructed a local Lagrange interpolation set for 𝑆²₇(▵_CT). Liu and Fan [8] constructed a local Lagrange interpolation set for 𝑆₅^2,3(▵_DCT), where ▵_DCT is the triangulation by refining some of triangles in ▵ with the double Clough-Tocher splits.
For general cases 𝑟≥1, a local Lagrange interpolation of 𝑆_𝑑^𝑟,𝜌(▵_CT) has been proposed by Nürnberger et al. in [9], where 𝑟, 𝜌, and 𝑑 are taken as the following related values ()()𝜌,𝑑∶=3𝑚+1,6𝑚+3,𝑟=2𝑚+1,3𝑚,6𝑚+1,𝑟=2𝑚, (4) with 𝑚=⌊𝑟/2⌋.
Let ◊ be a quadrangulation of Ω which consists of nondegenerate convex quadrilaterals. By adding one or two diagonals of each quadrilateral, some triangulated quadrangulations can be obtained. Nürnberger et al. [10] constructed a Lagrange interpolation scheme based on 𝐶¹ cubic splines on certain triangulations obtained from checkerboard quadrangulations. In [11], they also constructed a local Lagrange interpolation method based on 𝐶¹ cubic splines on certain triangulations obtained from a separable quadrangulation. Further they [12] described local Lagrange interpolation methods based on 𝐶¹ 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 𝐶² splines on any triangulated quadrangulation have not been developed. In this paper, we shall construct a Lagrange interpolation scheme on 𝑆₇^2,3() 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 𝒫,𝑆₇^2,3() 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 𝑇∶=⟨𝑣₁,𝑣₂,𝑣₃⟩ in Δ with vertices 𝑣₁,𝑣₂,𝑣₃, and the corresponding polynomial piece 𝑠∣_𝑇 is written in the form: 𝑠∣_𝑇=_{𝑖+𝑗+𝑘=𝑑}𝑐^𝑇_𝑖𝑗𝑘𝐵^𝑑_𝑖𝑗𝑘, (5) where 𝐵^𝑑_𝑖𝑗𝑘 are the Bernstein basis polynomials of degree 𝑑 associated with 𝑇. As usual, we identify the Bernstein-Bézier coefficients {𝑐^𝑇_𝑖𝑗𝑘}_{𝑖+𝑗+𝑘=𝑑} with the set of domain points 𝐷_𝑇∶={𝜉^𝑇_𝑖𝑗𝑘∶=(𝑖𝑣₁+𝑗𝑣₂+𝑘𝑣₃)/𝑑}_{𝑖+𝑗+𝑘=𝑑}. We write 𝐷_𝑑,Δ for the union of the sets of domain points associated with the triangles of Δ.

Given an integer 0≤𝑚≤𝑑, let 𝑅^𝑇_𝑚𝑣₁∶=𝜉^𝑇_𝑖𝑗𝑘,𝐷∶𝑖=𝑑−𝑚_𝑇𝑚𝑣₁∶=𝜉^𝑇_𝑖𝑗𝑘∶𝑖≥𝑑−𝑚. (6) We have similar definitions at the other vertices of 𝑇. If 𝑣 is a vertex of a triangulation ▵, we, respectively, define that the ring and the disk of radius 𝑚 around 𝑣 are the set 𝑅_𝑚(𝑣)∶=𝑅^𝑇_𝑚(𝑣),𝐷_𝑚(𝑣)∶=𝐷^𝑇_𝑚(𝑣), (7) where the union is taken over all triangles attached to 𝑣.

We recall [13] that supposes that 𝒮 is a subspace of 𝑆⁰_𝑑(Δ), and then 𝒫⊆𝐷_𝑑,Δ is said to be a determining set for 𝒮 that provides that, for any 𝑠∈𝒮,𝜆_𝜉𝑠=0 for all 𝜉∈𝒫 implies that 𝑠≡0. Note that 𝜆_𝜉 is a linear functional. The set 𝒫 is called a minimal determining set (MDS) for 𝒮 if there is no smaller determining set. Further, following [7], a basis {𝐵_𝑖}^𝑛_𝑖=1 for a spline space 𝒮 is called a stable local basis provided that constants ℓ,𝐾₁, and 𝐾₂ exist depending only on the smallest angle in ▵ such that(1)for each 1≤𝑖≤𝑛, there is a vertex 𝑣_𝑖 of ▵ for which supp(𝐵_𝑖)⊆star^ℓ(𝑣_𝑖),(2)for all choices of the coefficient vector 𝑐=(𝑐₁,…,𝑐_𝑛), 𝐾₁‖𝑐‖_∞≤^𝑛_𝑖=1𝑐_𝑖𝐵_𝑖∞≤𝐾₂‖𝑐‖_∞. (8)Here star¹(𝑣) is defined to be the set of all triangles surrounding vertex 𝑣, and star^ℓ(𝑣) is defined to be the union of the star¹(𝑤), where 𝑤 are vertices of star^ℓ−1(𝑣).

It is well known that a spline 𝑠 in 𝑆⁰_𝑑(Δ) is uniquely determined by its Bernstein-Bézier coefficient set {𝑐_𝜉}_𝜉∈𝐷_𝑑,Δ. In order to describe smoothness conditions for splines, we recall some notations introduced in [14]. Suppose that 𝑇∶=⟨𝑣₁,𝑣₂,𝑣₃⟩ and 𝑇∶=⟨𝑣₄,𝑣₃,𝑣₂⟩ are two adjoining triangles from Δ, which share the oriented edge 𝑒∶=⟨𝑣₂,𝑣₃⟩, and let 𝑠∣_𝑇=_{𝑖+𝑗+𝑘=𝑑}𝑐_𝑖𝑗𝑘𝐵^𝑑_𝑖𝑗𝑘,𝑠∣_𝑇=_{𝑖+𝑗+𝑘=𝑑}𝑐̃_𝑖𝑗𝑘𝐵^𝑑_𝑖𝑗𝑘, (9) where 𝐵^𝑑_𝑖𝑗𝑘 and 𝐵^𝑑_𝑖𝑗𝑘 are the Bernstein polynomials of degree 𝑑 on the triangles 𝑇 and 𝑇, respectively. Given integers 0≤𝑛≤𝑗≤𝑑, let 𝜏^𝑛_𝑗,𝑒 be linear functional defined on 𝑆⁰_𝑑(Δ) by 𝜏^𝑛_𝑗,𝑒𝑠∶=𝑐_{𝑛,𝑑−𝑗,𝑗−𝑛}−_{𝜈+𝜇+𝜅=𝑛}𝑐̃_{𝜈,𝜇+𝑗−𝑛,𝜅+𝑑−𝑗}𝐵^𝑛_𝜈𝜇𝜅𝑣₁. (10) These are called smoothness functionals of order 𝑛. According to [14], a spline 𝑠∈𝑆⁰_𝑑(Δ) belongs to 𝐶^𝑟(𝑇⋃𝑇) for some 𝑟>0 if and only if 𝜏^𝑛_𝑚,𝑒=0,𝑛≤𝑚≤𝑑,0≤𝑛≤𝑟. (11)

3. Checkerboard Triangulations

Definition 1 (see [10]). Suppose that ◊ is a quadrangulation consisting of quadrilaterals with largest interior angle less than 𝜋. 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 ℬ and 𝒲 denote the sets of black and white quadrilaterals of ◊, respectively. Following [10], throughout this paper, we also assume that all interior vertices of ◊ are of degree four. This assumption can ensure that there exists 𝒢⊂ℬ such that for every interior vertex 𝜈 of ◊, there is a unique quadrilateral 𝑄∈ℬ sharing the vertex 𝜈. For 𝑖=1,2,3,4, let 𝒲_𝑖 be the set of white quadrilaterals which share 𝑖 edges with black quadrilaterals. Let 𝑛_𝐵=#ℬ and 𝑛_𝑖∶=#𝒲_𝑖 for 𝑖=1,2,3,4, and let 𝑛_𝑉 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 𝒢 are shaded grey. It is noted that the other black quadrilaterals have not been colored.

Figure 1 

The set 𝒢 for a checkerboard triangulation.

4. Lagrange Minimal Determining Sets

Lemma 2 (see [10]). The set of all domain points in a triangle 𝑇 is a Lagrange minimal determining set for the space 𝑃_𝑑.

Let 𝑄∶=⟨𝑣₁,𝑣₂,𝑣₃,𝑣₄⟩ be a quadrilateral with vertices in counterclockwise order and be the triangulated quadrangulation of 𝑄 with 𝑣_𝑄 being the intersection point of the two diagonals of 𝑄. As shown in Figure 2, let 𝑇_𝑖∶=⟨𝑣_𝑄,𝑣_𝑖,𝑣_𝑖+1⟩ and 𝑒_𝑖∶=⟨𝑣_𝑖,𝑣_𝑄⟩ for 𝑖=1,2,3,4, where 𝑣₅=𝑣₁.

Figure 2 

The stable Lagrange MDS ℳ for 𝑆72,3() produced by Lemma 3.

Lemma 3. The set ℳ∶=²_𝑖=1𝐷^𝑇1₃𝑣_𝑖𝐷^𝑇1₃𝑣_𝑄𝐷^𝑇2₃𝑣₃𝐷^𝑇4₃𝑣₄𝜉^𝑇1₁₃₃,𝜉^𝑇1₂₃₂,𝜉^𝑇1₂₂₃,𝜉^𝑇1₃₃₁,𝜉^𝑇1₃₂₂,𝜉^𝑇1₃₁₃𝜉^𝑇2₁₃₃,𝜉^𝑇2₂₂₃,𝜉^𝑇2₃₁₃,𝜉^𝑇2₄₀₃𝜉^𝑇4₁₃₃,𝜉^𝑇4₂₃₂,𝜉^𝑇4₃₃₁,𝜉^𝑇4₄₃₀𝜉^𝑇3₁₃₃ (12) is a stable Lagrange MDS for 𝑆₇^2,3(). 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 𝑠∣_𝑇1 associated to 36 domain points in 𝑇₁ must be zero. By the 𝐶² smoothness conditions on the edge ⟨𝑣_𝑄,𝑣₂⟩ and the 𝐶³ smoothness conditions at vertex 𝑣₂, all Bernstein-Bézier coefficients of 𝑠∣_𝑇2 must be zero except for 14 coefficients 𝑐^𝑇2_𝑖𝑗𝑘 (𝑘=4,5,6,7,𝑖+𝑗=7−𝑘), 𝑐^𝑇2₄₀₃, 𝑐^𝑇2₃₁₃, 𝑐^𝑇2₂₂₃ and 𝑐^𝑇2₁₃₃. Let 𝐂∶=𝑐^𝑇2₀₀₇,𝑐^𝑇2₁₀₆,𝑐^𝑇2₀₁₆,𝑐^𝑇2₂₀₅,𝑐^𝑇2₁₁₅,𝑐^𝑇2₀₂₅,𝑐^𝑇2₃₀₄,𝑐^𝑇2₂₁₄,𝑐^𝑇2₁₂₄,𝑐^𝑇2₀₃₄,𝑐^𝑇2₄₀₃,𝑐^𝑇2₃₁₃,𝑐^𝑇2₂₂₃,𝑐^𝑇2₁₃₃^𝑇, (13) then the homogenous Lagrange interpolation conditions at the associated 14 domain points lead to 𝐆𝐂=𝟎, (14) where𝐆=^{610000000000000777667603×6576005×64760005×6376600077706676003×65760005×6476500007772×5676012×55760040×547600080×53765000777567656763×55766×55763×55765×547615×54765×54765×547620×537620×537630×537620×5376577702×56760012×557600040×5476400007773×4676027×457600135×4476000405×437640007}₇₇^2×46₇₆⁴⁶₇₆^12×45₇₆^12×45₇₆^3×45₇₆^40×44₇₆^60×44₇₆^30×44₇₆^5×44₇₆^80×43₇₆^160×43₇₆^160×43₇₆^40×43_{76477746762×46763×457612×457612×45765×447630×447660×447640×44765×437640×437660×4376160×4376477703×46760027×4576000135×44763000077721×3676048×357600320×34760001280×337630007773×3676367627×357618×35763×3576135×347627×34769×34765×3476405×3376540×3376270×337660×337637772×36762×367612×357624×357612×357640×3476130×3476130×347640×347680×3376320×3376480×3376320×3376377736763×36763×357618×357627×35765×347645×3476135×3476135×34765×337660×3376270×3376540×3376}. (15)
It is easy to see that rank(𝐆)=14. Thus all the remaining 14 Bernstein-Bézier coefficients of 𝑠∣_𝑇2 are zero. Similarly, all the Bernstein-Bézier coefficients of 𝑠∣_𝑇4 must be also zero by smoothness conditions and given Lagrange interpolation conditions. Further, using 𝐶² smoothness conditions along two edges ⟨𝑣_𝑄,𝑣₃⟩ and ⟨𝑣_𝑄,𝑣₄⟩ and 𝐶³ smoothness conditions at vertices 𝑣₃ and 𝑣₄, all Bernstein-Bézier coefficients of 𝑠∣_𝑇3 are zero except for 𝑐^𝑇3₁₃₃. Using the interpolation condition at domain point 𝜉^𝑇3₁₃₃: 𝐵⁷₁₃₃𝑐^𝑇3₁₃₃=0, that is; 𝑐^𝑇3₁₃₃=0 since 𝐵⁷₁₃₃=14580/117649. Therefore all B coefficients of 𝑠∣_𝑇3 are zero.
The construction in Lemma 3 is stable in the sense that the maximum coefficient of 𝑠 is bounded by 𝐾max_𝜉∈ℳ|𝑝(𝜉)|, where 𝐾 is a constant depending only on the smallest angle in ▵. Thus, we say that ℳ is a stable Lagrange MDS for 𝑠∈𝒮₇^2,3(). The proof is completed.

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

Lemma 4. Suppose that  consists of two triangulated quadrangulations 𝑄₁=⟨𝑣₁,𝑣₂,𝑣₃,𝑣₄⟩ and 𝑄₂=⟨𝑣₃,𝑣₅,𝑣₆,𝑣₄⟩ sharing the edge ⟨𝑣₃,𝑣₄⟩, where 𝑄₁∈ℬ and 𝑄₂∈𝒲. Let 𝑣_𝑄1 and 𝑣_𝑄2 be the points where the two diagonals of 𝑄₁ and 𝑄₂ intersect, respectively. Let the set ℳ∶=ℳ_𝑄1𝑚₁𝑚₂𝑚₃, (16) where ℳ_𝑄1 is the set defined in Lemma 3 for the triangulated quadrangulations of 𝑄₁ and 𝑚₁=𝐷^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃𝑣_𝑄2𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₂₂,𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₁₃,𝑚₂=𝐷^{⟨𝑣𝑄2,𝑣3,𝑣5⟩}₃𝑣₅𝜉^{⟨𝑣𝑄2,𝑣3,𝑣5⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣5⟩}₂₂₃,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣5⟩}₃₁₃,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣5⟩}₄₀₃,𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₁₃₃,𝑚₃=𝐷^{⟨𝑣𝑄2,𝑣6,𝑣4⟩}₃𝑣₆𝜉^{⟨𝑣𝑄2,𝑣6,𝑣4⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣4⟩}₂₃₂,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣4⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣4⟩}₄₃₀, (18) be a stable Lagrange MDS for 𝒮₇^2,3(). These domain points are marked with • in Figure 3.

Figure 3 

The point set ℳ produced by Lemma 4.

Proof. Using Lemma 3, we can see that all of the B coefficients of 𝑠 associated with domain points in 𝑄₁ are uniquely determined by the data. Writing 𝑠∣_{⟨𝑣𝑄2,𝑣4,𝑣3⟩} in B form and using 𝐶¹, 𝐶², and 𝐶³ smoothness conditions, all B coefficients are determined except for the B coefficients associated with 𝑚₁. Because 𝐆𝐂=𝑏, where 𝑏∶=𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₇₀₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₆₁₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₆₀₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₅₂₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₅₁₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₅₀₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₄₃₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₄₂₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₄₁₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₄₀₃,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₃₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₂₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣4,𝑣3⟩}₃₁₃^𝑇,𝐆=^{61000000000000777667603×6576005×6476600000077706676003×65760005×647650007772×5676012×55760040×54765000000777567656763×55766×55763×55765×547615×54765×54765×547620×537620×537630×5376577702×56760012×557600040×547640007773×4676027×457600135×4476}₄⁰⁰⁰⁰⁰⁰_{7772×4676467612×457612×45763×457640×447660×447630×44765×4476160×4376160×437640×4376477746762×46763×457612×457612×45765×447630×447660×447640×447640×437660×4376160×4376477703×46760027×4576000135×447630007773×3676367627×357618×35763×3576135×347627×34769×34765×3476540×3376270×337660×337637772×36762×367612×357624×357612×357640×3476130×3476130×347640×3476320×3376480×3376320×3376377736763×36763×357618×357627×35765×347645×3476135×3476135×347660×3376270×3376540×3376}. (19)
It is easy to see that rank(𝐆)=13. Thus, all of the remaining B coefficients of 𝑚₁ are uniquely determined by the data. Then, using Lemma 3 again, it is easy to see that all of the B coefficients of 𝑠 associated with domain points in 𝑄₂ are uniquely and stable determined by the data. Therefore, ℳ is a stable Lagrange MDS for 𝒮₇^2,3(). The proof is completed.

Lemma 5. Suppose that consists of three triangulated quadrangulations 𝑄₁, 𝑄₂, and 𝑄₃ as in Figure 4, where 𝑄₁,𝑄₃∈ℬ and 𝑄₂∈𝒲. Let the set ℳ∶=ℳ_𝑄1ℳ_𝑄3𝑚₁𝑚₂, (20) where ℳ_𝑄𝑖 is the set defined in Lemma 3 for the triangulated quadrangulations of 𝑄_𝑖 and 𝑚₁=𝐷^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃𝑣_𝑄2𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₂₂,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₁₃,𝑚₂=𝜉^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₄₃₀,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃, (21) be a stable Lagrange MDS for 𝒮₇^2,3(). These domain points are marked with • in Figure 4.

Figure 4 

The point set ℳ produced by Lemma 5.

Proof. Using Lemma 3, we can see that all of the B coefficients of 𝑠 associated with domain points in 𝑄₁ and 𝑄₃ are uniquely determined by the data. Arguing as in Lemma 4, all the B coefficients of 𝑠∣_{⟨𝑣𝑄2,𝑣3,𝑣2⟩} associated with domain points are uniquely determined by the data too. Writing 𝑠∣_{⟨𝑣𝑄2,𝑣2,𝑣5⟩} in B form and using 𝐶¹, 𝐶² and 𝐶³ smoothness conditions, all B coefficients are determined except for 𝑚∶={𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₂₂₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₁₃₃}. The Lagrange interpolation and 𝐶¹ and 𝐶² smoothness conditions at 𝑒∶=⟨𝑣_𝑄2,𝑣₅⟩ imply that these coefficients must satisfy the linear system 𝐆𝐂=𝑏, where 𝐂∶=𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₂₂₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₁₃₃,𝑐^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₃₃₁^𝑇,𝑏∶=𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₁₃₃,0,0^𝑇,𝐆=^{1280×337600005×377620×367630×357620×347605×337660×3376270×3376540×33760}_{𝛾1−100𝛽1𝛾210−102𝛽1𝛾1}, (22) where 𝑣₂=𝛽₁𝜈₆+𝛾₁𝜈_𝑄2. The determinant of the matrix 𝐆 is −5×10³𝛽₁8𝛾₁+9. (23) By the geometric meaning of the 𝛽₁ and 𝛾₁, we have 𝛽₁<0 and 𝛾₁>0. Thus det(𝐆)≠0. So, all B coefficients of 𝑠∣_{⟨𝑣𝑄2,𝑣2,𝑣5⟩} are uniquely determined. By the same principle, all B coefficients of 𝑠∣_{⟨𝑣𝑄2,𝑣6,𝑣3⟩} are uniquely determined. Using Lemma 3 again, it is easy to see that all B coefficients of 𝑄₂ are uniquely and stable determined by the data. Therefore ℳ is a stable Lagrange MDS for 𝒮₇^2,3(). The proof is completed.

Lemma 6. Suppose that consists of three triangulated quadrangulations 𝑄₁, 𝑄₂, and 𝑄₃ as in Figure 5, where 𝑄₁,𝑄₃∈ℬ and 𝑄₂∈𝒲. Let the set ℳ∶=ℳ_𝑄1ℳ_𝑄3𝑚₁𝑚₂−𝐷^{⟨𝑣𝑄2,𝑣2,𝑣7⟩}₃𝑣₂, (24) where ℳ_𝑄𝑖 is the set defined in Lemma 3 for the triangulated quadrangulations of 𝑄_𝑖 and 𝑚₁=𝜉^{⟨𝑣𝑄2,𝑣2,𝑣3⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃,𝑚₂=𝐷^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₃𝑣₆𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₂₂₃,𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₃₁₃,𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₄₀₃, (25) be a stable Lagrange MDS for 𝒮₇^2,3(). These domain points are marked with • in Figure 5.

Figure 5 

The point set ℳ produced by Lemma 6.

Proof. Using Lemma 3 and 𝐶¹, 𝐶², and 𝐶³ smoothness conditions, we can see that all of the B coefficients of 𝑠 associated with domain points in 𝑄₁ and 𝑄₃ are uniquely determined by the data. Writing 𝑠∣_{⟨𝑣𝑄2,𝑣3,𝑣6⟩} in B form and using 𝐶¹, 𝐶², and 𝐶³ smoothness conditions, all B coefficients are determined except for the B coefficients of 𝑚∶=⋃𝐷^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃(𝑣_𝑄2)⋃{𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₂₂,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₁₃} associated. The Lagrange interpolation and 𝐶¹ and 𝐶² smoothness conditions at 𝑒∶=⟨𝑣_𝑄2,𝑣₃⟩ and 𝑒∶=⟨𝑣_𝑄2,𝑣₂⟩ imply that these coefficients must satisfy the linear system 𝐆𝐂=𝑏, where 𝐂∶=𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₇₀₀,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₆₁₀,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₆₀₁,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₂₀,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₁₁,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₀₂,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₃₀,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₂₁,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₁₂,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₀₃,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₃₁,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₂₂,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₃₁^𝑇,𝑏∶=𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₇₀₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₆₁₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₆₀₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₂₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₁₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₅₀₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₃₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₂₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₁₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₀₃,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₃₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₂₂,0,0,0^𝑇,𝐆=^{6100000000000000777667603×6576005×647660000000077706676003×65760005×64765000007772×5676012×55760040×5476500000000777567656763×55766×55763×55765×547615×54765×54765×547620×537620×537630×537650077702×56760012×557600040×54764000007773×4676027×457600135×44764000000007}₇₇^2×46₇₆⁴⁶₇₆^12×45₇₆^12×45₇₆^3×45₇₆^40×44₇₆^60×44₇₆^30×44₇₆^5×44₇₆^160×43₇₆^160×43₇₆^40×43_{7640077746762×46763×457612×457612×45765×447630×447660×447640×447640×437660×4376160×437640077703×46760027×4576000135×44763000007773×3676367627×357618×35763×3576135×347627×34769×34765×3476540×3376270×337660×33763007772×36762×367612×357624×357612×357640×3476130×3476130×347640×3476320×3376480×3376320×337600000000𝛾1000𝛽100−10000000000𝛾200−10𝛽2000000000𝛾2200002𝛽2𝛾2}, (26) where 𝜈₆=𝛽₁𝜈₂+𝛾₁𝜈_𝑄2 and 𝜈₃=𝛽₂𝜈₅+𝛾₂𝜈_𝑄2. The determinant of the matrix G is −7.07759×10⁻¹⁰𝛽₂𝛾₂. (27) By the geometric meaning of the 𝛽₂ and 𝛾₂, we have 𝛽₂<0 and 𝛾₂>0. Thus det(𝐆)≠0. So, all B coefficients of 𝑠∣_{⟨𝑣𝑄2,𝑣3,𝑣2⟩} are uniquely determined. Applying Lemmas 3 and 4, it is easy to see that all B coefficients of 𝑄₂ are uniquely and stable determined by the data. Therefore ℳ is a stable Lagrange MDS for 𝒮₇^2,3(). The proof is completed.

Lemma 7. Suppose that consists of three triangulated quadrangulations 𝑄₁, 𝑄₂, 𝑄₃, and 𝑄₄ as in Figure 6, where 𝑄₁,𝑄₃,𝑄₄∈ℬ and 𝑄₂∈𝒲. Let the set ℳ∶=ℳ_𝑄1ℳ_𝑄3ℳ_𝑄4𝑚₁−𝐷^{⟨𝑣𝑄4,𝑣2,𝑣9⟩}₃𝑣₂𝐷^{⟨𝑣𝑄4,𝑣10,𝑣5⟩}₃𝑣₅, (28) where ℳ_𝑄𝑖 is the set defined in Lemma 3 for the triangulated quadrangulations of 𝑄_𝑖 and 𝑚₁=𝐷^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃𝑣_𝑄2𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₂₂,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₁₃₃,𝜉^{⟨𝑣𝑄2,𝑣6,𝑣3⟩}₄₃₀,𝜉^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₃₁₃,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₄₃₀,𝜉^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₃₁, (29) be a stable Lagrange MDS for 𝒮₇^2,3(). These domain points are marked with • in Figure 6.

Figure 6 

The point set ℳ produced by Lemma 7.

Proof. Using Lemma 3 and 𝐶¹, 𝐶², and 𝐶³ smoothness conditions, we can see that all of the B coefficients of 𝑠 associated with domain points in 𝑄₁, 𝑄₃, and 𝑄₄ are uniquely determined by the data. Writing 𝑠∣_{⟨𝑣𝑄2,𝑣2,𝑣5⟩} in B form and using 𝐶¹, 𝐶², and 𝐶³ smoothness conditions, all B-coefficients are determined except for the B-coefficients of 𝑚∶=⋃𝐷^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃(𝑣_𝑄2)⋃{𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₃₁,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₂₂,𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃} associated. The Lagrange interpolation and 𝐶¹, 𝐶² smoothness conditions at 𝑒∶=⟨𝑣_𝑄2,𝑣₂⟩ and 𝑒∶=⟨𝑣_𝑄2,𝑣₅⟩ imply that these coefficients must satisfy the linear system 𝐆𝐂=𝑏, where 𝐂∶=𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₇₀₀,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₆₁₀,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₆₀₁,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₂₀,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₁₁,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₀₂,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₃₀,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₂₁,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₁₂,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₃₁,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₂₂,𝑐^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣3,𝑣2⟩}₃₁₃,𝑐^{⟨𝑣𝑄2,𝑣5,𝑣6⟩}₃₃₁^𝑇,𝑏∶=𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₇₀₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₆₁₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₆₀₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₂₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₁₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₅₀₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₃₀,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₂₁,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₁₂,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₄₀₃,𝑝𝜉^{⟨𝑣𝑄2,𝑣2,𝑣5⟩}₃₂₂,0,0,0,0^𝑇,𝐆=^{6100000000000000777667603×6576005×647660000000077706676003×65760005×64765000007772×5676012×55760040×5476500000000777567656763×55766×55763×55765×547615×54765×54765×547620×537620×537630×537650077702×56760012×557600040×54764000007773×4676027×457600135×44764000000007772×4676467612×457612×45763×457640×447660×447630×44765×4476160×4376160×437640×4376}₄⁰⁰_{77746762×46763×457612×457612×45765×447630×447660×447640×447640×437660×4376160×437640077703×46760027×4576000135×44763000007772×36762×367612×357624×357612×357640×3476130×3476130×347640×3476320×3376480×3376320×337600000000𝛾2000𝛽200−10000000𝛾220002𝛽2𝛾20000000000000𝛾100−10𝛽10000000002𝛾2100002𝛽1𝛾1}, (30)
where 𝑣₂=𝛽₁𝑣₆+𝛾₁𝑣_𝑄2, 𝑣₃=𝛽₂𝑣₅+𝛾₂𝑣_𝑄2. The determinant of the matrix G is −1.31659×10⁻⁸𝛽₁𝛽₂𝛾₁𝛾₂. (31) By the geometric meaning of the 𝛽₁, 𝛽₂, 𝛾₁, and 𝛾₂, we have 𝛽₁<0, 𝛽₂<0, 𝛾₁>0, and 𝛾₂>0. Thus det(𝐆)≠0. So, all B coefficients of 𝑠∣_{⟨𝑣𝑄2,𝑣3,𝑣2⟩} are uniquely determined. Applying Lemmas 3 and 4, it is easy to see that all B coefficients of 𝑄₂ are uniquely and stable determined by the data. Therefore, ℳ is a stable Lagrange MDS for 𝒮₇^2,3(). The proof is completed.

5. Construction of a Lagrange Interpolation Pair

Theorem 8. Suppose that is checkerboard triangulation. Then (32) Moreover, the following set ℳ of domain points is a stable Lagrange MDS:(1)if 𝑄∈𝒢, choose points as in Lemma 3,(2)if 𝑄∈ℬ⧵𝒢, choose points as in Lemma 3, leaving out the points in the sets 𝐷^𝑇₃ whenever 𝑣 is a vertex of 𝑄 which is interior to ◊,(3)suppose that 𝑄∶=⟨𝑣₁,𝑣₂,𝑣₃,𝑣₄⟩∈𝒲, and let 𝑒_𝑖∶=⟨𝑣_𝑖,𝑣_𝑖+1⟩ for 𝑖=1,2,3,4, where 𝑣₅=𝑣₁ and 𝑣_𝑄 is the point where the two diagonals of 𝑄 intersect:(a)if 𝑄∈𝒲₁, choose points as in Lemma 4,(b)if 𝑄∈𝒲₂, choose points as in Lemmas 5 and 6,(c)if 𝑄∈𝒲₃, choose points as in Lemma 7,(d)if 𝑄 sharing four edges with black quadrilaterals, say 𝑒₁,𝑒₂,𝑒₃,and𝑒₄, choose the points 𝜉^𝑇1₂₂₃, 𝜉^𝑇2₄₀₃, 𝜉^𝑇3₄₀₃, and 𝐷^𝑇1₃(𝑣_𝑄).

Proof. To establish that ℳ is a Lagrange MDS, suppose 𝒮₇^2,3() and that we are given values for 𝑠(𝜉) for all 𝜉∈ℳ. We need to show that all of the B coefficients of 𝑠 are uniquely determined. By Lemma 3, all B-coefficient 𝑠 associated with domain points lying in quadrilaterals 𝑄∈𝒢 is uniquely determined. Now consider 𝑄∈ℬ⧵𝒢. For each vertex 𝑣∈𝑄 which is an interior vertex of ◊, the B coefficients corresponding to domain points in the disk 𝐷^𝑇₃ are already uniquely determined by 𝐶¹,𝐶², and 𝐶³ 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 𝑠 corresponding to the remaining domain points in 𝑄 are uniquely determined.
Now suppose that 𝑄∈𝒲. If 𝑄∈𝒲₁, then, using Lemma 4, we can see that all B coefficients of 𝑠 corresponding to the remaining domain points in 𝑄 are uniquely determined. If 𝑄∈𝒲₂, then, using Lemmas 5 and 6, we can see that all B-coefficients of 𝑠 corresponding to the remaining domain points in 𝑄 are uniquely determined. If 𝑄∈𝒲₃, then using Lemma 7, we can see that all B coefficients of 𝑠 corresponding to the remaining domain points in 𝑄 are uniquely determined. If 𝑄 shares four edges with black quadrilaterals, we can then argue exactly as in Lemma 7 to see that all B coefficients of 𝑠 corresponding to the remaining domain points in 𝑄 are uniquely determined. Since we have shown that ℳ is a MDS, it follows that dim𝒮₇^2,3()=#ℳ.
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 Δ. This follows from that the computations of Lemmas 2–7 are stable, and the fact that computing coefficients from 𝐶¹,𝐶², and 𝐶³ smoothness conditions is automatically stable.

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

Figure 7 

The Lagrange MDS produced by Theorem 8.

Theorem 9. Let be a checkerboard triangulation, and let ℳ be the set defined in Theorem 8. Then there exists a stable local basis {𝐵_𝑖}^𝑛_𝑖=1 of 𝒮₇^2,3(), where 𝑛∶=#ℳ.

Proof. It is clear that if ℳ is a MDS for 𝒮₇^2,3(), then, for each 𝜉∈ℳ, there exists a unique spline 𝐵_𝜉∈𝑆, satisfying 𝜆_𝜂𝐵_𝜉=𝛿_𝜉,𝜂, for all 𝜂∈ℳ. The splines 𝐵_𝜉 obviously form a basis for 𝒮₇^2,3(), which is commonly called the dual basis corresponding to ℳ.
The proof of Theorem 8 shows that all B coefficients of 𝐵_𝜉 are uniquely and stable determined. So we should give the support of 𝐵_𝜉. As in [15], suppose that 𝜉 lies in a quadrilateral 𝑄. Then we claim that(1)supp(𝐵_𝜉)=𝑄 if 𝜉=𝒲,(2)supp(𝐵_𝜉)⊂star(𝑄_𝜉) if 𝜉∈ℬ⧵𝒢,(3)supp(𝐵_𝜉)⊂star²(𝑄_𝜉) otherwise.

We are now ready to discuss interpolation. Suppose that is a checkerboard triangulation, and that 𝐵_𝜉 are the dual basis function of Theorem 9 corresponding to Lagrange MDS for 𝒮₇^2,3() defined in Theorem 8. Then for every 𝑓∈𝐶(Ω), there is a unique spline 𝑠=ℐ𝑓∈𝒮 which satisfies 𝑠(𝜂)=ℐ𝑓(𝜂)=𝑓(𝜂), for all 𝜂∈ℳ. This defines a linear projector ℐ mapping 𝐶(Ω) onto 𝒮. We now give an error bound for this interpolation method.

Theorem 10. Suppose 𝑓 lies in the Sobolev space 𝑊_∞^𝑚+1(Ω) for some 0≤𝑚≤7. Then 𝐷^𝛼_𝑥𝐷^𝛽_𝑦𝑓−ℐ𝑓_Ω≤𝐾|Δ|^{𝑚+1−𝛼−𝛽}𝑓_𝑚+1,Ω (33) for 0≤𝛼+𝛽≤𝑚. Here |•|_𝑚+1,Ω is usual Sobolev seminorm, and |Δ| is the maximum of the diameters of the triangles in Δ. The constant 𝐾 depends only on the smallest angle in Δ.

Proof. It was shown in Section10 of [16] that if a space of splines 𝒮 of degree 𝑑 contains 𝒫_𝑑 and has a stable local basis, then it provides optimal order approximations of smooth functions.

The result of Theorem 10 can also be established with the weak-interpolation methods described in [17].

Acknowledgments

The authors would like to thank Josefa Linares-Perez and the other anonymous referees whose comments substantially enhanced the quality of the paper. This work subsidized by the National Natural Science Foundation of China under Grant 10871100 and the Innovation Project for Graduate Education of Jiangsu Province under Grant CXLX11_0865.