Approximation power of polynomial splines on T-meshes

Computer Aided Geometric Design 29 (2012) 599–612

Contents lists available at SciVerse ScienceDirect

Computer Aided Geometric Design www.elsevier.com/locate/cagd

Approximation power of polynomial splines on T-meshes ✩ Larry L. Schumaker ∗ , Lujun Wang Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States

a r t i c l e

i n f o

Article history: Received 17 November 2011 Received in revised form 12 April 2012 Accepted 23 April 2012 Available online 27 April 2012

a b s t r a c t Polynomial spline spaces defined on T-meshes are useful tools for both surface modeling and the finite element method. Here the approximation power of such spline spaces is established. The approach uses Bernstein–Bézier methods to get precise conditions on the geometry of the meshes which lead to local and stable bases. © 2012 Elsevier B.V. All rights reserved.

Keywords: Splines T-meshes T-nodes Hanging vertices Approximation power

1. Introduction Polynomial splines defined on T-meshes were introduced in 2006 in the paper Deng et al. (2006), and were further studied in a series of papers Deng et al. (2006, 2008), Huang et al. (2006a, 2006b), Jin et al. (2009), Li and Chen (2011), Li et al. (2006, 2007, 2009, 2010), Mourrain (2010). They have been applied in surface fitting (Li et al., 2007), and for approximating signed distance functions (Song et al., 2010). They are closely related to T-splines, see Remark 1, which are very useful for curve and surface design, see e.g. Sederberg et al. (2003, 2004). In addition, they have also attracted attention for use in the finite element method, see e.g. Dörfel et al. (2010), Nguyen-Thanh et al. (2011), Tian et al. (2011) and references therein. Polynomial splines on T-meshes are a generalization of tensor product splines. Tensor product spline spaces have stable local bases, and they provide optimal order approximation of smooth functions. Our main goal in this paper is to show that polynomial splines on T-meshes have these same properties. Our analysis of splines on T-meshes will be based on Bernstein–Bézier methods. They are the key tool in the study of splines on triangulations, see Lai and Schumaker (2007), and were first used for splines on T-meshes in Deng et al. (2006). The paper is organized as follows. In Section 2 we introduce the T-meshes of interest here, and discuss various properties of them. In Section 3 we define our spline spaces, and provide a variety of notation and background material needed later. To make the paper self contained, and since it requires very little additional effort, in Section 4 we give a compact derivation of the well-known dimension formulae, along with a construction of a natural set of basis functions. In Section 5 we discuss the support properties of these basis functions, and in Section 6 we treat their stability. Section 7 contains our main results on approximation power, which are established for an explicit quasi-interpolation operator based on averaged tensor product polynomials. We conclude the paper with some remarks and references. ✩

*

This paper has been recommended for acceptance by B. Juettler. Corresponding author. E-mail addresses: [email protected] (L.L. Schumaker), [email protected] (L. Wang).

0167-8396/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cagd.2012.04.003

600

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Fig. 1. Some regular T-meshes.

Fig. 2. T-meshes with holes.

Fig. 3. T-meshes with cycles.

2. T-meshes 2.1. Basics



Definition 2.1. Let  := { R i }iN=1 be a collection of axis-aligned rectangles such that the interior of the domain Ω := R i is connected. In addition, suppose that any pair of distinct rectangles R i , R j can intersect each other only at points on their edges. Then we call  a T-mesh. A T-mesh  forms a partition of the domain Ω . The domain Ω need not be rectangular. Moreover, it may have one or more holes. T-meshes include tensor product meshes as a special case. However, in contrast to tensor product meshes, T-meshes are allowed to have T-nodes, where a T-node is a vertex of one rectangle that lies in the interior of an edge of another rectangle. We also refer to these as hanging vertices, cf. Schumaker and Wang (2011, in press). Note that T-nodes can be either in the interior or on the boundary of Ω . If v is a T-node of a T-mesh , then we refer to the unique rectangle R v in  such that v lies in the interior of an edge of R v as the hanging rectangle associated with v. Figs. 1–3 show several examples of T-meshes. Definition 2.2. We say that a T-mesh  is regular provided that for every vertex v of , the set of all rectangles containing v has a connected interior. The mesh in Fig. 2 (right) is not regular. It is important to clarify what we mean by an edge of a T-mesh. If e :=  v , w  is a line segment of  connecting two vertices v, w of  such that there are no vertices lying in the interior of e, then we call e an edge segment. If e :=  v , w  is a line segment of  connecting two vertices v, w of  such that all vertices lying in the interior of e are T-nodes, and if e cannot be extended to a longer line segment with the same property, then we say that e is a composite edge of . A composite edge can consist of one or more edge segments. Note that some composite edges are edges of rectangles in , but not all edges of rectangles are composite edges. 2.2. Cycles and refinement We recall the following definition from Schumaker and Wang (2011). Definition 2.3. Suppose w 1 , . . . , w n is a collection of T-nodes in a T-mesh  such that for each i = 1, . . . , n, the vertex w i lies in the interior of a composite edge with one endpoint at w i +1 , where we set w n+1 = w 1 . Then we say that w 1 , . . . , w n form a cycle.

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

601

The T-mesh shown in Fig. 3 (left) has a cycle of four boundary T-nodes, while the one in Fig. 3 (right) has a cycle of four interior T-nodes. In this paper we will restrict ourselves to regular T-meshes that have no cycles. This is not a major restriction for applications since, as we show in Theorem 2.4 below, if we start with a T-mesh without any cycles and refine it in a natural way, then no cycles will be introduced. The natural way to refine a T-mesh is as follows: Split Algorithm: Choose a rectangle R, and insert an edge segment e (parallel to the x or y axis) to split R into two subrectangles. The two subrectangles need not be the same size, and either or both of the endpoints of e may be ˜. hanging vertices in the new T-mesh  This process can be repeated as often as needed to locally refine a given T-mesh.

˜ is a T-mesh obtained by repeatedly splitting rectangles into two Theorem 2.4. Suppose  is a T-mesh with no cycles, and that  ˜ subrectangles. Then  also has no cycles. Proof. The proof of this result is precisely the same as the proof of the analogous result for triangulations with hanging vertices, see Theorem 2.6 of Schumaker and Wang (in press), and there is no need to repeat it here. It is based on an assignment of a binary tree to each hanging vertex. 2 2.3. Some notation For later use we introduce some notation. Given a regular T-mesh  without cycles, let V NT = the number of vertices of  that are not T-nodes, E hor = the number of horizontal composite edges of , E ver = the number of vertical composite edges of , E c = E hor + E ver = the number of composite edges in , N = the number of rectangles in . 3. Polynomial splines on a T-mesh Suppose  is a T-mesh, and let 0  r1 < d1 and 0  r2 < d2 be given integers. Let r = (r1 , r2 ) and d = (d1 , d2 ). Then the associated space of splines of degree d and smoothness r is defined to be the finite dimensional linear space

  Sdr () := s ∈ C r (Ω): s| R i ∈ Pd for all i = 1, . . . , N ,

where Pd :=

d ,d span{xi y j }i =1 0,2j =0

(3.1)

is the usual space of tensor product polynomials of degree d1 in x and degree d2 in y. j

Here C (Ω) denotes the space of functions s such that their mixed derivatives D ix D y s are continuous for all 0  i  r1 and 0  j  r2 . r

3.1. The Bernstein–Bézier representation of a tensor product polynomial In this and the next several subsections we introduce notation and describe a framework for analyzing the approximation properties of Sdr (). The main tool is the Bernstein–Bézier representation, which was used already in Deng et al. (2006) for getting dimension formula for splines on T-meshes. Let R := [x1 , x2 ] × [ y 1 , y 2 ] be a rectangle, and let p be a tensor d d d d2 product polynomial in Pd . Suppose { B i 1 (x)}i =1 0 and { B j 2 ( y )} j = are the univariate Bernstein polynomials associated with 0 the intervals [x1 , x2 ] and [ y 1 , y 2 ], respectively. Then we can represent p in Bernstein–Bézier form relative to R as

p (x, y ) =

d1 d2  

d

d

c iRj B i 1 (x) B j 2 ( y ).

(3.2)

i =0 j =0

Let

 ξiRj :=

 (d1 − i )x1 + ix2 (d2 − j ) y 1 + j y 2 , , d1

d2

(3.3)

for 0  i  d1 and 0  j  d2 . We call these the domain points associated with R, and set

 d ,d Dd, R := ξiRj i =1 0,2j =0 .

(3.4)

To get a single subscript notation for the Bernstein–Bézier representation of p, let d

d

B ξR (x, y ) := B i 1 (x) B j 2 ( y ),

all ξ = ξiRj in Dd, R .

(3.5)

602

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Then we can rewrite (3.2) as

p (x, y ) =



c ξR B ξR (x, y ).

(3.6)

ξ ∈Dd, R

We call (3.6) the Bernstein–Bézier representation of p, and refer to the c ξR as the B-coefficients. μ ,μ R Let v 1 = (x1 , y 1 ), and let μ = (μ1 , μ2 ). Then we call the set of domain points D μ ( v 1 ) := {ξiRj }i =10, j2=0 the disk of size

ξiRj

μ with 0  i  ν

around v 1 . Disks around the other three vertices of R can be defined similarly. Domain points of the form are said to lie within a distance ν of the left edge e of R. We write this as d(ξ, e )  ν , and define the analogous notation for the other three edges of R. 3.2. Stability of the B-form

Let m = (d1 + 1)(d2 + 1), and let { g 1 , . . . , gm } be the tensor product Bernstein basis polynomials in (3.5) arranged in lexicographical order. Suppose {ζ1 , . . . , ζm } are the points in Dd, R arranged in the same order, and let c := (c 1 , . . . , cm ) T be the corresponding vector of coefficients in (3.6). We write c ∞ for the max-norm of c. We also write  ·  R for the sup-norm of a function defined on R, and  · q, R for the corresponding q-norm. Theorem 3.1. There exists a constant K 1 depending only on d such that

c ∞ K1

  p  R  c ∞ .

Proof. The upper bound is obvious in view of the fact that the basis functions in (3.6) are nonnegative and sum to one. is For the lower bound, we note that by well-known results on tensor product interpolation, the matrix M := [ g j (ζi )]m i , j =1 nonsingular. Now suppose p is a polynomial in Pd written in the B-form (3.6) relative to a rectangle R. Then Mc = r, where r = [ p (ζ1 ), . . . , p (ζm )] T . But then

    c ∞   M −1 r ∞   M −1 ∞ r ∞ ,

and since r ∞   p  R , the result follows with K 1 =  M −1 ∞ .

2

Using equivalence of norms on finite dimensional spaces, we can also establish stability in the q-norm for any 1  q < ∞, cf. Theorem 2.7 of Lai and Schumaker (2007) for the analogous case of total degree polynomials written in B-form relative to a triangle. Theorem 3.2. Given a rectangle R, let A R be its area. Then there exists a constant K 2 depending only on d such that 1/q

AR

K2

1/q

c q   p q, R  A R c q .

3.3. The Bernstein–Bézier representation for piecewise polynomials Given a T-mesh , we define the associated set of domain points to be

Dd, :=



Dd, R ,

(3.7)

R ∈

where here the union is to be understood in the sense that multiple appearances of the same point are allowed. For an arbitrary T-mesh , the set Dd, is in one-to-one correspondence with the space PP d () of piecewise tensor product polynomials of degree d defined over , with no continuity required between pieces. Since Sdr () is a subspace of PP d (), every spline s ∈ Sdr () can be associated with a unique set of B-coefficients. However, not every set of numbers {c ξ }ξ ∈Dd, can serve as the coefficients of a spline in Sdr () – they must be constrained by certain smoothness conditions to be discussed in more detail in the following section. Given μ = (μ1 , μ2 ) and a vertex v of a T-mesh, we define

D μ ( v ) :=



R Dμ ( v ),

(3.8)

R ∈ v

where as in (3.7) we allow multiple appearances of the same point in the union. Here  v is the set of all rectangles in  that have a vertex at v. If e is a composite edge of  and ξ is a domain point in some rectangle with an edge e˜ lying on e, then we write d(ξ, e )  ν provided d(ξ, e˜ )  ν .

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

603

3.4. Smoothness conditions Here and throughout the remainder of the paper, whenever we list the vertices of a rectangle we assume they are in counter-clockwise order. We need the following lemma later. Lemma 3.3. Let s be a spline in Sdr (), and let v be a vertex of . Let R :=  v , v 2 , v 3 , v 4  and R˜ =  v , v 5 , v 6 , v 7  be two rectangles are uniquely determined that share the vertex v. Suppose we are given the coefficients {c ξ }ξ ∈ D R ( v ) . Then the coefficients {c η } R˜ r

by the C r smoothness at v.

η∈ D r ( v )

Proof. We give the proof for the case where  v , v 2  is a horizontal edge of R – the proof when it is vertical is similar. r ,r

˜

r ,r

Suppose {c i j }i ,1j =20 and {˜c i j }i ,1j =20 are the B-coefficients corresponding to the domain points in D rR ( v ) and D rR ( v ), respectively. Let D 1 and D 2 be the directional derivatives associated with the vectors v 2 − v and v 4 − v, respectively. Then for all 0  i  d1 and 0  j  d2 , j

D 1i D 2 s| R ( v ) =

d1 ! d2 ! j i  c 00 , (d1 − i )! (d2 − j )! 1 2

(3.9)

j

where 1i and 2 denote the i-th and j-th forward difference operators in the first and second indices, respectively. Similarly, for all 0  i  d1 and 0  j  d2 , j

D 3i D 4 s| R˜ ( v ) =

d1 ! d2 ! j i  c˜ 00 , (d1 − i )! (d2 − j )! 1 2

(3.10)

where D 3 and D 4 are the directional derivatives corresponding to the vectors v 5 − v and v 7 − v. There exist nonzero constants a1 , a2 such that

D 3 = a1 D 2 ,

D 4 = a2 D 1 ,

or

D 3 = a1 D 1 ,

D 4 = a2 D 2 ,

depending on whether R and R˜ share an edge or not. Equating the formulae for the derivatives of s| R and s| R˜ at v for all r1 ,r2 0  i  r1 and 0  j  r2 , we get (r1 + 1)(r2 + 1) equations which can be solved for the coefficients {˜c i j }i = in terms 0, j =0 r ,r

1 2 . If we write the equations in groups according to the degree of the derivatives involved, the of the coefficients {c i j }i = 0, j =0 matrix corresponding to this system is lower triangular with nonzero entries on the diagonal. 2

3.5. Minimal determining sets Minimal determining sets play an important role in the theory of splines on triangulations, see Lai and Schumaker (2007). This concept can be extended to work with splines on T-meshes, see e.g. Deng et al. (2006). Let S () be a subspace of PP d (). Definition 3.4. Suppose M is a subset of Dd, such that for any spline s ∈ S (),

c ξ = 0,

all ξ ∈ M

implies

s ≡ 0,

(3.11)

where for any ξ ∈ M, c ξ is the corresponding B-coefficient of s. Then we call M a determining set for S (). If there is no smaller set with this property, then we call M a minimal determining set for S (). It follows from simple linear algebra that if M is a determining set for S (), then dim S ()  #M, see Section 5.6 of Lai and Schumaker (2007). Moreover, if M is a minimal determining set for S (), then dim S () = #M. 3.6. Splines on composite edges In this section we discuss the nature of a spline and its cross derivatives along a composite edge e. Given a composite edge e of , let



re =

r1 , if e is vertical, r2 , if e is horizontal,

De = and

de =

D x, D y,

d2 , d1 ,

if e is vertical, if e is horizontal,

if e is vertical, if e is horizontal.

(3.12) (3.13)

(3.14)

604

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Fig. 4. The rectangles in the proof of Lemma 3.6.

Lemma 3.5. Let s ∈ Sdr (), and suppose e is a composite edge of . Then for each 0  ν  re , D eν s|e is a univariate polynomial of degree de . Proof. First consider ν = 0. The assertion is trivial if e contains at most one T-node in its interior, so we may suppose e :=  v 0 , v m+1  contains T-nodes v 1 , . . . , v m in its interior with m  2. For each i = 1, . . . , m, let R i be the hanging rectangle associated with v i , and let p i := s| R i . Now for each 2  i  m, the polynomials p i −1 and p i coincide on the interval [ v i −1 , v i ], and thus p i −1 ≡ p i on e. It follows that p 1 ≡ p 2 ≡ · · · ≡ pm on e, which establishes the result for ν = 0. The proof for 1  ν  re is a minor modification. 2 The following lemma shows how the polynomials D eν s| R are determined from B-coefficients associated with domain points near the edge e. Given a composite edge e := u , v  of , let R e := u 1 , u 2 , u 3 , u 4  be a rectangle in  such that u 1 = u. In addition, let Rˆ e :=  v 1 , v 2 , v 3 , v 4  be a rectangle in  with v 1 = v. We consider the case where  v 4 , v 1  and u 1 , u 2  both lie on e, see Fig. 4. The other possible cases are similar. If e does not contain any T-nodes in its interior, then we may take R e = Rˆ e . Let

⎧  r ,d −r −1 ⎨ ξiRj e 1 2 2 , if e is vertical, i =0, j =r2 +1 Me :=   ⎩ ξ R e d1 −r1 −1,r2 , if e is horizontal. i j i =r +1, j =0

(3.15)

1

R

R

These are just the domain points ξ ∈ Dd, R e with d(ξ, e )  re that lie outside the disks D r e (u 1 ) and D r e (u 2 ). Lemma 3.6. Let e := u , v  be a composite edge of , and suppose we are given the B-coefficients of a spline s ∈ Sdr () associated with all domain points in

˜ e := D rR e (u ) ∪ D rRˆ e ( v ) ∪ Me . M

(3.16)

Then all coefficients of s associated with domain points ξ with d(ξ, e )  re are uniquely determined. Proof. Let p be the polynomial that agrees with s on R e . We examine the case where e is horizontal. The case where it is j vertical is similar. By Lemma 3.5, p and its derivatives D e up to order re at any point on e agree with those of s. Let D 1 , D 2 , D 3 , D 4 denote the directional derivatives associated with the vectors u 2 − u 1 , u 4 − u 1 , v 4 − v 1 , and v 2 − v 1 , respectively. Then using (3.9), we can compute



j

r1 ,r2

j

d1 −r1 −1,r2

D 1i D 2 p (u ) i =0, j =0 ,



j

r1 ,r2

D 3i D 4 p ( v ) i =0, j =0 ,

(3.17)

and



D 1i D 2 p (u ) i =r +1, j =0 1

(3.18)

˜ e. from the B-coefficients of p associated with domain points in M ˜ 1 be the intersection Now suppose R :=  w 1 , w 2 , w 3 , w 4  is a rectangle in  such that the edge  w 3 , w 4  lies on e. Let w ˜ 2 be the intersection of the line through v 2 , of the line through u 4 , u 1 with the line through w 1 , w 2 . Similarly, let w ˜ 1, w ˜ 2 , v 1 , u 1 , see Fig. 4. Let D˜ 1 , D˜ 2 , D˜ 3 , D˜ 4 denote the directional v 1 with the line through w 1 , w 2 , and let R˜ :=  w ˜ 1 − u 1 , u 1 − v 1 , and w ˜ 2 − v 1 , respectively. Then there exist constants such derivatives associated with the vectors v 1 − u 1 , w that

˜ i = ai D i , D

i = 1, . . . , 4.

(3.19)

˜ i . Then we use Given (3.17) and (3.18), we can compute the values of the same sets of derivatives with D i replaced by D these to compute the B-coefficients of p relative to R˜ corresponding to all domain points in Dd, R˜ with d(ξ, e )  re . Finally, we can get the coefficients of p relative to R corresponding to domain points ξ ∈ Dd, R with d(ξ, e )  re by subdivision of ˜ 2 the rectangle R.

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

605

4. A basis for Sdr () Suppose  is a regular T-mesh without cycles, and let d  2r + 1, i.e., d1  2r1 + 1 and d2  2r2 + 1. In this section we construct a convenient basis for Sdr (), while at the same time rederiving a known formula for its dimension. To accomplish this, we construct a natural minimal determining set for Sdr (). Let VNT and E be the sets of non-T vertices and composite edges of , respectively. For each v ∈ VNT , let R v be a rectangle with an edge e v with an endpoint at v such that all other edges with an endpoint at v have lengths at most |e v |. R Let M v be the disk D r v ( v ). For each composite edge e of , let Me be the set defined in (3.15). Finally, for each rectangle R in , let

 d −r −1,d −r −1 M R := ξiRj i =1 r +1 1,r +2 1 2 . 1

(4.1)

2

Now let

M :=

v ∈VNT

Mv ∪



Me ∪

e ∈E



MR .

(4.2)

R ∈

Lemma 4.1. The set M is a determining set for Sdr (), i.e., (3.11) holds. Proof. Suppose s ∈ Sdr () and that c ξ = 0 for all ξ ∈ M. Let v ∈ VNT . Then using Lemma 3.3 we see that all B-coefficients of s associated with domain points in D r ( v ) are zero. Now if e is a composite edge whose ends are both in VNT , then by Lemma 3.6 it follows that c ξ = 0 for all domain points with d(ξ, e )  re . We now proceed iteratively. Mark all of the vertices and composite edges treated so far as determined. Then repeat the following steps until all composite edges have been dealt with: 1) For each T-node v on a determined composite edge, use Lemma 3.6 to conclude that all B-coefficients of s associated with domain points in the disk D r ( v ) are zero. Mark these vertices as determined. 2) For each composite edge e := u , v  such that both u and v are determined, use Lemma 3.6 and the fact that M contains Me to show that all B-coefficients of s associated with domain points ξ with d(ξ, e )  re must be zero. Mark these edges as determined. Since  does not contain any cycles, this process stops only when all vertices and edges are marked, which means that all B-coefficients of s corresponding to domain points within a distance re of any edge e of  must be zero. To complete the proof that M is a determining set, we now show that the remaining B-coefficients are also zero. These are the coefficients corresponding to domain points whose distance to any edge e of  is greater then re . But these are just the domain points in the sets M R , and by the definition of M, we conclude that these coefficients are also zero. We have shown that s ≡ 0, and the proof of the lemma is complete. 2 For any domain point ξ ∈ Dd, and any spline s ∈ Sdr (), we define γξ s = c ξ , where c ξ is the B-coefficient of s associated with ξ . Lemma 4.2. For each ξ ∈ M there exists a unique spline ψξ ∈ Sdr () such that

γη ψξ = δξ,η , all η ∈ M.

(4.3)

Proof. Fix ξ ∈ M and set c ξ = 1 and c η = 0 for all other η ∈ M. We then determine the polynomial pieces of ψξ stepwise exactly as in the proof of Lemma 4.1, except that now we start with one nonzero coefficient. In carrying out this process, no inconsistencies arise in computing the B-coefficients of ψξ since the disks of size r around the vertices are separated. This is why we have assumed that d  2r + 1. 2 Our next theorem makes use of the notation introduced in Section 2.3. Theorem 4.3. The set M is a minimal determining set for Sdr (), and

dim Sdr () = (r1 + 1)(r2 + 1) V NT + (r2 + 1)(d1 − 2r1 − 1) E hor

+ (r1 + 1)(d2 − 2r2 − 1) E ver + (d1 − 2r1 − 1)(d2 − 2r2 − 1) N .

(4.4)

Moreover, the set Ψ := {ψξ }ξ ∈M is a basis for Sdr (). Proof. As observed in Section 3.5, the fact that M is a determining set for Sdr () implies dim Sdr ()  #M. On the other hand, the splines in Ψ are clearly linearly independent in view of the dual property (4.3), and thus dim Sdr ()  #M. We

606

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Fig. 5. A T-mesh with some basis functions with large support and others with very small support.

conclude that M is a minimal determining set, dim Sdr () = #M, and Ψ is a basis for Sdr (). To complete the proof, we observe that the cardinality of M is given by the formula in (4.4). 2 Corollary 4.4. Suppose  is a regular T-mesh without cycles, and let d = (d, d) and r = (r , r ) with d  2r + 1. Then

dim Sdr () = (r + 1)2 V NT + (r + 1)(d − 2r − 1) E c + (d − 2r − 1)2 N .

(4.5)

5. Support properties of the basis Ψ In this section we show how the support properties of the basis Ψ for Sdr () depend on the nature of the T-mesh . We begin with two definitions. Definition 5.1. For every composite edge e := u , v  of , we define star(e ) to be the set of rectangles that have an edge overlapping with e. Definition 5.2. Let e be a composite edge of , and suppose e 1 , . . . , em is a maximal sequence of composite edges such that for each i = 1, . . . , m, one end of e i is in the interior of e i +1 , where em+1 = e. We call e 1 , . . . , em a chain ending at e. We refer to m as the length of the chain. We illustrate this concept in Fig. 5. There is one chain of length three ending on the horizontal composite edge with right end point at the open dot. In addition, there are five chains of length one ending on the horizontal composite edge with left end point at the open dot, and five more ending on the vertical composite edge with top end point at the open dot. For each composite edge e of , let βe be the length of the longest chain ending at e. To state a result on the supports of the basis functions in Ψ , we need some additional notation. Let e be a composite edge of . If e does not contain any T-nodes in its interior, then there are no chains ending at e, and we define Ae = {e }. If there are m T-nodes in the interior of e, then there are at least m chains ending on e. In this case we define Ae to be the union of e together with all composite edges that lie on these chains. For any vertex v, let A v be the union of all Ae corresponding to composite edges e ending at v. Theorem 5.3. Let {ψξ }ξ ∈M be the dual basis for Sdr () defined in Lemma 4.2. Then for each ξ ∈ M, the support σ (ψξ ) of ψξ satisfies

⎧ ⎨ e∈A v star(˜e ), if ξ ∈ M v for some vertex v ∈ VNT , σ (ψξ ) ⊆ star(˜e ), if ξ ∈ Me for some composite edge e , ⎩ e˜ ∈Ae R, if ξ ∈ M R for some rectangle R .

Proof. Suppose ξ ∈ M v for some vertex v ∈ VNT . Let R be a rectangle of  with the property that none of its four edges is contained in a chain ending at a composite edge with a vertex at v. Then for each edge e of R, ψξ and all of its cross derivatives up to order re vanish at all points on e. Since γη ψξ = 0 for all η ∈ M R , it follows that ψξ | R ≡ 0. Now suppose ξ ∈ Me for some edge e of . Let R be a rectangle of  with the property that none of its four edges is contained in a chain ending at e. Then for every edge e˜ of R, ψξ and all of its cross derivatives up to order re˜ vanish at all points of e˜ . Since γη ψξ = 0 for all η ∈ M R , it follows that ψξ | R ≡ 0. Finally, suppose ξ ∈ M R for some rectangle R of . Then for every edge e of R, ψξ and all of its cross derivatives up to order re vanish on e, and it follows that ψξ vanishes except on R. 2 Clearly, once we allow T-nodes, the supports of some of the basis functions can become large. For example, let r = (0, 0) and d = (1, 1) and consider the T-mesh in Fig. 5. Then the basis function corresponding to the point ξ ∈ M marked with an open circle has support on the entire square minus the shaded region. On the other hand, if ξ ∈ M lies in the shaded region in Fig. 5, then the corresponding basis function ψξ has small support. Thus, if we choose  with some care, we

ˆ have small supports. This can ensure that on some parts Ωˆ of the domain, all basis functions with support overlapping Ω means that we will get good local approximation properties.

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

607

6. Stability of the MDS M and the basis Ψ To state our results on stability, we need some additional notation. For every composite edge e consisting of m edge segments e 1 , . . . , em with m  1, let



αe := max

 |e | |e | , , |e 1 | |em |

and let βe be the length of the longest chain ending on e. For each rectangle R in , let longest edge to the length of its shortest edge, and let

α := max αe , e ∈E

β := max βe ,

κ R be the ratio of the length of its

κ := max κ R ,

(6.1)

R ∈

e ∈E

where we recall that E is the set of all composite edges of . We now show that the minimal determining set M for Sdr () is stable. Theorem 6.1. For any s ∈ Sdr (), its associated B-coefficients satisfy

|c η |  K 3 max |c ξ |, ξ ∈M

η ∈ Dd, ,

(6.2)

where K 3 is a constant depending only on d, α , β , and κ . Proof. Suppose we are given the B-coefficients {c ξ }ξ ∈M of a spline s ∈ Sdr (). Then following the steps in the proof of Lemma 4.1, we can compute all of the other B-coefficients of s. We claim that all of these computed coefficients are bounded as in (6.2). R We begin by showing that (6.2) holds if η lies in a disk D r ( v ) around a vertex v ∈ VNT . Recall that M v = D r v ( v ), where R v is a rectangle with an edge e v with one endpoint at v and such that all other edges with an endpoint at v have lengths at most |e v |. Now for any other rectangle R˜ with a vertex at v, we apply Lemma 3.3 to compute the coefficients c η of s ˜

corresponding to all domain points η in D rR ( v ). The constants a1 , a2 appearing in the proof of that lemma are bounded by a constant depending only on κ . Since derivatives up to at most max(r1 , r2 ) are involved, it follows that

|c η |  C 1 max |c ξ |, ξ ∈M v

˜

η ∈ D rR ( v ),

where C 1 depends only on d and κ . Next we examine the computation in Lemma 3.6 associated with a composite edge e. We make use of the notation in ˜ e be as in (3.16), and let p be the polynomial that agrees with s on R e . Then that lemma and in Fig. 4. In particular, let M by the formula in (3.9), it follows that all of the derivatives in (3.17) and (3.18) are bounded by C 2 maxξ ∈M ˜ |c ξ |, where C 2 e

˜i is a constant depending only on d. Using these, we can compute the analogous sets of derivatives with D i replaced by D for i = 1, . . . , 4. The constants in (3.19) are bounded by a constant depending on d, α and κ . From these derivatives we compute the coefficients of p associated with domain points η ∈ R˜ with d(η, e )  re . These coefficients are bounded by C 3 maxξ ∈M ˜ |c ξ |, where C 3 is a constant depending only on d, α and κ . The coefficients corresponding to domain points e

in any subrectangle of R˜ are obtained by subdivision, and can only be smaller. To finish the proof, we have to carry out the iterative process described in Steps 1)–2) of the proof of Lemma 4.1. The coefficients in Step 1) are associated with domain points in D r ( v ), where v is a T-node lying on a composite edge e. These were already determined when we computed the coefficients corresponding to domain points with d(ξ, e )  re . Bounds for Step 2) follow as above. However, as we carry out these steps, the constants can multiply. But the number of times this can happen is limited by the size of β . 2 For any rectangle R, the associated tensor product Bernstein polynomials are nonnegative and sum to one. This means that they are bounded by one at all points in R, and Theorem 6.1 implies

sΩ  K 3 max |c ξ |, ξ ∈M

where K 3 is the constant in (6.2). Theorem 6.2. The basis Ψ is stable in the sense that

ψξ   K 3 ,

for all ξ ∈ M,

(6.3)

where K 3 is the constant in (6.2). Proof. Fix ξ ∈ M, and consider ψξ . Then for all

η ∈ M, we have cη = δξ,η , and (6.3) follows immediately from (6.2). 2

608

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

7. Approximation power of Sdr () In this section we show that Sdr () has optimal approximation order for smooth functions. 7.1. Averaged tensor product Taylor polynomials Averaged tensor product Taylor polynomials are a useful tool for obtaining error bounds for tensor product polynomial approximation, see e.g. Section 4.6 of Brenner and Scott (1994). Here we review the main facts. Given a disk B in R2 , let  g B be a function in C 0∞ (R2 ) with support on B and with B g B = 1. For the construction of such a function, see Section 4.1 of Brenner and Scott (1994). Fix d = (d1 , d2 ). Given any function f ∈ L 1 ( B ), let

F d, B f (x, y ) =

 d1 d2    1 j (−1)i + j f (u , v ) D ui D v (x − u )i ( y − v ) j g B (u , v ) du dv . i! j! i =0 j =0

(7.1)

B

Then F d, B f belongs to Pd , and is called the averaged tensor product Taylor polynomial associated with f . It has several useful properties, see Section 4.6 of Brenner and Scott (1994). For example, if p ∈ Pd , then F d, B p = p. Now suppose A is a domain in R2 , and let | A | be its diameter, i.e., the diameter of the smallest disk containing A. Then we define the shape parameter of A as

κˆ A := where

| A|

ρA

(7.2)

,

ρ A is the radius of the largest disk B contained in A. Now if f ∈ L q ( B ) with 1  q  ∞, then  F d, B f q, A  K 4  f q, B ,

(7.3)

where K 4 is a constant depending only on d and κˆ A , see Brenner and Scott (1994). Given m and 1  q  ∞, let be the corresponding classical Sobolev space. The following result corresponds to Theorem 4.6.11 in Brenner and Scott (1994).

W qm ( A )

Theorem 7.1. Suppose 0  m1  d1 and 0  m2  d2 and let m = min(m1 , m2 ). Suppose f ∈ W qm+1 ( A ) for some 1  q  ∞. Then for all ν , μ  0, with ν + μ  m,

  m +1    ν μ  1 +1   D D y ( f − F d, B f )  K 5 | A |m+1−ν −μ  D m f q, A +  D y 2 f q, A , x x q, A

(7.4)

where the constant K 5 depends only on d and the shape parameter κˆ A . 7.2. A quasi-interpolant Let Ψ = {ψξ }ξ ∈M be the dual basis for Sdr () in Theorem 4.3. For each domain point ξ ∈ M, let R ξ be a rectangle in  such that ξ ∈ Dd, R ξ . Given f ∈ L 1 (Ω), let F d, B ξ f be the averaged tensor product polynomial of degree d associated with the largest disk B ξ contained in the rectangle R ξ . Now define

Q f :=



γξ ( F d, B ξ f )ψξ ,

(7.5)

ξ ∈M

γξ are the linear functionals appearing in Lemma 4.2 that pick off B-coefficients. For each rectangle R in , let   Γ R := ξ ∈ M: σ (ψξ ) ∩ R = ∅ , (7.6)

where the

where

σ (ψξ ) is the support of ψξ . Let Ω R := σ (ψξ ).

(7.7)

ξ ∈Γ R

Theorem 7.2. The operator Q is a linear projector mapping L 1 (Ω) onto Sdr (). Moreover, there exists a constant K 6 such that for all 1  q  ∞, all rectangles R ∈ , and all f ∈ L q (Ω),

 Q f q, R  K 6  f q,Ω R , where K 6 depends only on d, α , β , and κ .

(7.8)

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

609

Proof. It is clear from the definition that Q is a linear operator. The claim that Q s = s for all splines s ∈ Sdr () follows from the fact that, as observed above, for any ξ ∈ M, F d, B ξ p = p for all tensor product polynomials p ∈ Pd . We now establish (7.8) in the case 1  q < ∞. The case q = ∞ is similar and simpler. Given ξ ∈ M, let R ξ be the rectangle used in constructing Q , and let A R ξ be its area. Then by Theorem 3.2 and (7.3),

  K2 K2 K4 |c ξ | = γξ ( F d, B ξ f )  1/q  F d, B ξ f q, R ξ  1/q  f q, B ξ , A Rξ

A Rξ

where K 2 and K 4 both depend only on d. Now fix a rectangle R in . Then the stability of the minimal determining set M as expressed in (6.2) implies

|c η | 

K2 K3 K4 1/q

A min

 f q,Ω R ,

η ∈ Dd, R ,

where A min is the area of the rectangle in Ω R with smallest area, and K 3 is the constant in (6.2). Here Dd, R is the set of domain points in R defined in (3.4). Using the nonnegativity of the Bernstein basis polynomials and the fact that they form a partition of unity, this immediately implies

 Q f q, R =

q 1/q    1/q   AR R  c B  K K K η η 2 3 4 1/q  f q,Ω R ,  R

A min

η∈Dd, R

where A R is the area of R. Now it is easy to see that A R  C 1 A min , where C 1 depends only on this, we get (7.8). 2

α , β , and κ . Inserting

7.3. Approximation power of Sdr () We first establish a local approximation result. Theorem 7.3. Suppose R is a rectangle in a T-mesh , and let | R | be its diameter. Let Ω R be the corresponding cluster of rectangles defined in (7.7). Let 0  m1  d1 and 0  m2  d2 and define m = min(m1 , m2 ). Suppose f ∈ W qm+1 (Ω R ) for some 1  q  ∞. Then for all ν , μ  0 with ν + μ  m,

 ν μ   m +1   1 +1    D D y ( f − Q f )  K 7 | R |m+1−ν −μ  D m f q,Ω +  D y 2 f q,Ω . x x q, R R

R

(7.9)

The constant K 7 depends only on d, α , β , and κ . Proof. Let A be the convex hull of Ω R , and let p := F d, B f be the averaged tensor product Taylor polynomial associated with the largest disk B that is contained in A. By Theorem 1.8 in Lai and Schumaker (2007), f can be extended from Ω R to A in such a way that the norm of the derivatives of f on A are bounded by a constant C 1 times the norm of the derivatives on Ω R . This constant depends on the Lipschitz constant of the boundary of A, which can be bounded by an absolute constant since the angles at the vertices of A are all greater than ninety degrees. Since Q reproduces polynomials in Pd ,

    ν μ  ν μ     D D y ( f − Q f )   D ν D μ  x x y ( f − p ) q, R + D x D y Q ( f − p ) q, R . q, R

Now since Q ( f − p ) is a polynomial, using the Markov inequality (see e.g. Cheney, 1966), and then applying (7.8), we have

 ν μ   D D y Q ( f − p)   x q, R

C2

| R |ν + μ

   Q ( f − p )  C 2 K 6  f − p q,Ω , R q, R | R |ν + μ

where C 2 depends only on d. It is not hard to see that there exists a constant C 3 depending only on α , β , and κ , such that | A | = |Ω R |  C 3 | R | and κˆ A  C 3 κ , where κˆ A is the shape parameter associated with A. Then using (7.4), we get (7.9). 2 It is straightforward to extend this to a global version for functions f ∈ W qm+1 (Ω) with Ω R replaced by Ω , and with | R | replaced by the mesh size of  defined by || = max R ∈ | R |. 8. Remarks Remark 1. T-spline spaces were introduced and studied in Sederberg et al. (2003, 2004). They are defined on a T-mesh associated with a rectangle, and are linear spaces spanned by certain collections of NURBS defined on the mesh. As such, on each rectangle R of the mesh a T-spline is a rational function of the form p / w, where both p and w are tensor product polynomials. If the denominator is taken to be constant (say one), then a T-spline reduces to a polynomial spline on the T-mesh. In Sederberg et al. (2003) these were called standard T-splines. T-splines were originally invented as a tool for CAGD purposes, but more recently have been also used in the isogeometric approach to solving PDE’s, see e.g. Dörfel et al. (2010).

610

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Remark 2. Most of the listed papers on polynomial splines on T-meshes assume that the underlying domain is a rectangle. A number of the papers further restrict themselves to so-called hierarchical T-meshes. Unfortunately, there does not seem to be agreement on what a hierarchical T-mesh is. In Deng et al. (2008), Jin et al. (2009), Li et al. (2007), Nguyen-Thanh et al. (2011) they are defined as T-meshes obtained by starting with a rectangle (or in some cases a tensor product mesh), and applying a splitting algorithm which splits a given rectangle into four subrectangles. On the other hand, in Mourrain (2010) a hierarchical T-mesh is defined to be the result of refining a given rectangle by repeatedly dividing rectangles into two subrectangles. Clearly, every 4-hierarchical T-mesh is automatically 2-hierarchical. Theorem 2.4 shows that every such hierarchical T-mesh is cycle free. Remark 3. T-meshes defined on a rectangular domain have been called regular T-meshes in the literature. We prefer to use the word regular for T-meshes with the property described in Definition 2.2. This is more in line with standard terminology in the theory of splines on triangulations, see Section 4.3 in Lai and Schumaker (2007). Remark 4. Bicubic splines defined on 4-hierarchical T-meshes were called PHT-splines in Deng et al. (2008), see also Tian et al. (2011) where they are used in the finite element method for elliptic equations. They are of course special cases of the splines considered here. Remark 5. The analog of T-nodes in the case of triangulations are called hanging vertices. Our recent paper Schumaker and Wang (in press) contains a detailed study of polynomial splines on triangulations with hanging vertices. Remark 6. In our earlier paper Schumaker and Wang (2011), we dealt with C 0 splines on TR-meshes consisting of a mix of triangles and rectangles with sides not necessarily parallel to the Cartesian coordinate axes. TR-meshes contain both T-meshes and triangulations with hanging nodes as special cases. Explicit minimal determining sets and the corresponding dual basis splines were used to establish approximation power of the spline spaces. To get results for meshes with cycles, we had to restrict ourselves to C 0 splines. Remark 7. It is easy to show by induction that if  is a regular T-mesh, then

E hor = 2N − E hi , E ver = 2N − i

E iv ,

V NT = V + 4N − 2

(8.1)



E hi

+

E iv

(8.2)



,

(8.3)

where V i is the number of interior vertices in , and E hi and E iv are the number of interior horizontal and vertical edge segments in , respectively. These formula are analogs of the well-known Euler relations for regular ordinary triangulations, see Section 4.4 of Lai and Schumaker (2007). It is easy to given an example to show that (8.3) does not hold for T-meshes that are not regular. Remark 8. Using the Euler relations of the previous remark, it is easy to see that the formula (4.4) for the dimension of Sdr () on a regular T-mesh can be rewritten as

dim Sdr () = (d1 + 1)(d2 + 1) N − (r2 + 1)(d1 + 1) E hi − (r1 + 1)(d2 + 1) E iv + (r1 + 1)(r2 + 1) V i .

(8.4)

This is the formula given in some of the earlier papers, see e.g. Deng et al. (2006), Huang et al. (2006a). Remark 9. The problem of finding the dimension of polynomial spline spaces defined on T-meshes has recently been studied using methods from homological algebra, see Mourrain (2010). In some cases it is even possible to get explicit dimension results for the case d < 2r + 1, albeit with some restrictions on the mesh. The results of Li and Chen (2011) show that for d < 2r + 1 it is not always possible to get explicit formulae for all regular cycle-free T-meshes. In particular, in that paper 1, 1 they give an example of regular cycle-free T-mesh such that dimension of the spline space S2,2 () depends on the spacing of the grid lines. This is analogous to what happens with polynomial splines on triangulations, where for low values of the polynomial degree compared to the smoothness, the dimension is also unstable and depends on the exact geometry of the triangulation, see Example 9.13 in Lai and Schumaker (2007). Remark 10. The dimension of polynomial spline spaces on T-meshes can be explored with the java program of Peter Alfeld, see www.math.utah.edu/~pa. His software can also be used to choose minimal determining sets for a given spline space. Remark 11. A simple argument shows that the basis functions Ψ for Sdr () given in Theorem 4.3 form a partition of unity, but are not necessarily nonnegative. When r1 = r2 = 0, the basis functions are also nonnegative, see Schumaker and Wang (2011).

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

611

Remark 12. Using certain tensor product B-splines, it is possible to construct a different basis for Sdr () which consists of nonnegative functions forming a partition of unity, see e.g. Deng et al. (2008). This basis may be more useful for modeling purposes. Remark 13. There is a version of Theorem 7.1 for approximation with averaged Taylor polynomials which gives more precise m +m +1 information on the order of convergence, but uses higher derivatives on the right-hand side. Suppose f ∈ W q 1 2 ( A ) with 0  m1  d1 and 0  m2  d2 for some 1  q  ∞. Then for all 0  ν  m1 and 0  μ  m2 ,

 ν μ     1 +1 −ν  m 1 +1 μ  m +1 −μ  2 +1   D D y ( f − F d, B f )  K 8 hm D x D ν Dm D y f q, A + h y 2 f q, A , x x x y q, A

(8.5)

where the constant K 8 depends only d and the shape parameter κˆ A . See Theorem 13.20 in Schumaker (2007). Remark 14. The existence of a polynomial satisfying (8.5) in the case ν = μ = 0 is shown in Dahmen et al. (1980), but the proof is not constructive. The case of derivatives is mentioned, but not treated explicitly. Remark 15. To establish the error bound (8.5) in the case where for any k = (k1 , k2 ),



j

ν = μ = 0, it is enough to assume that f ∈ W (m1 +1,m2 +1) ( A ), 

W qk ( A ) := f : D ix f ∈ L q ( A ) for 0  i  k1 and D y f ∈ L q ( A ) for 0  j  k2 .

(8.6)

It is shown in Dahmen et al. (1980) that if f ∈ W qk (Ω), then f also belongs the classical Sobolev space W qk (Ω) for all k < min(k1 , k2 ). Remark 16. The problem of finding error bounds for interpolation with tensor product polynomials on a rectangle was studied recently in Mößner and Reif (2009). Both Lagrange and Hermite interpolation are studied. There it is shown that the ordinary tensor product Taylor polynomial satisfies a bound similar to (8.5), but with an extra term on the right involving the norm of a mixed derivative. They give an explicit example where the ordinary Taylor polynomial does not satisfy a bound of the form (8.5). Remark 17. Our analysis of the stability of the dual basis splines in Theorem 4.3 shows that although refining a T-mesh by splitting rectangles into two allows for very flexible refinement algorithms, this should be done with some care to control the constants α , β , and κ that enter into our approximation results. Remark 18. For applications, there are two approaches to computing with Sdr (). One is to work with the full set of coefficients defining a piecewise polynomial in PP d (), and then enforce the smoothness conditions as a set of linear side conditions. The second approach is to make use of the minimal determining set M. As shown in Schumaker (2008) for the case of splines on ordinary triangulations, this leads to much smaller linear systems when doing interpolation or solving finite element problems. Remark 19. It is straightforward to construct a nodal minimal determining set and a corresponding Hermite interpolation method based on Sdr (). We explore error bounds for this interpolant in a separate paper, see Schumaker and Wang (2011). References Brenner, S.C., Scott, L.R., 1994. The Mathematical Theory of Finite Element Methods. Springer, New York. Cheney, E.W., 1966. Introduction to Approximation Theory. McGraw–Hill, New York. Dahmen, W., DeVore, R., Scherer, K., 1980. Multidimensional spline approximations. SIAM J. Numer. Anal. 17, 380–402. Deng, J.-S., Chen, F.-L., Feng, Y.-Y., 2006. Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194, 267–283. Deng, J.-S., Chen, F.-L., Li, X., Hu, C., Tong, W., Yang, Z., Feng, Y., 2008. Polynomial splines over hierarchical T-meshes. Graphical Models 70, 76–86. Dörfel, M.R., Juettler, B., Simeon, B., 2010. Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Engrg. 199, 264–275. Huang, Z.-J., Deng, J.-S., Li, X., 2006a. Dimensions of spline spaces over general T-meshes. J. Univ. Sci. Technol. China 36, 573–581. Huang, Z.-J., Deng, J.-S., Feng, Y., Chen, F., 2006b. New proof of dimension formulae spline spaces over general T-meshes via smoothing cofactors. J. Comput. Appl. Math. 24, 501–514. Jin, L., Deng, J., Chen, F., 2009. Submesh splines over hierarchical T-meshes. Int. J. CAD/CAM 9, 47–53. Lai, M.J., Schumaker, L.L., 2007. Spline Functions on Triangulations. Cambridge University Press, Cambridge. Li, Xin, Chen, Falai, 2011. On the instability in the dimension of splines spaces over T-meshes. Comput. Aided Geom. Design 28, 420–426. Li, C.-J., Wang, R.-H., Zhang, F., 2006. Improvement on the dimensions of spline spaces on T-mesh. J. Inform. Comput. Sci. 3, 235–244. Li, X., Deng, J., Chen, F., 2007. Surface modeling with polynomial splines over hierarchical T-meshes. Vis. Comput. 23, 1027–1033. Li, X., Deng, J., Chen, F., 2009. C 1 bicubic splines over general T-meshes. In: IEEE Conf. on Comp.-Aided Design and Comp. Graphics, pp. 93–95. Li, X., Deng, J., Chen, F., 2010. Polynomial splines over general T-meshes. Vis. Comput. 26, 277–286. Mößner, B., Reif, U., 2009. Error bounds for polynomial tensor product interpolation. Computing 86, 185–197. Mourrain, B., 2010. On the dimension of spline spaces on planar T-subdivisions. Preprint, arXiv:1011.1752v1.

612

L.L. Schumaker, L. Wang / Computer Aided Geometric Design 29 (2012) 599–612

Nguyen-Thanh, N., Nguyen-Xuan, H., Borda, S.P.A., Rabczuk, T., 2011. Isogeometric analysis using polynomial splines over hierarchical T-meshes for twodimensional elastic solids. Comput. Methods Appl. Engrg. 200, 1892–1908. Schumaker, L.L., 2007. Spline Functions: Basic Theory, third ed. Cambridge University Press, Cambridge. Schumaker, L.L., 2008. Computing bivariate splines in scattered data fitting and the finite element method. Numer. Algorithms 48, 237–260. Schumaker, L.L., Wang, Lujun, 2011. Spline spaces on TR-meshes with hanging vertices. Numer. Math. 118, 531–548. Schumaker, L.L., Wang, Lujun, 2011. On Hermite interpolation with polynomial splines on T-meshes. Submitted for publication. Schumaker, L.L., Wang, Lujun, in press. Splines on triangulations with hanging vertices. Constr. Approx. Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A., 2003. T-splines and T-NURCCSs. ACM Trans. Graph. 22, 477–484. Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T., 2004. T-spline simplification and local refinement. ACM Trans. Graph. 23, 276– 283. Song, X., Jüttler, B., Poteaux, A., 2010. Hierarchical spline approximation of the signed distance function. In: IEEE Int. Conf. on Shape Modeling and Appl., pp. 241–245. Tian, L., Chen, F., Du, Q., 2011. Adaptive finite element methods for elliptic equations over hierarchical T-meshes. J. Comput. Appl. Math. 236, 878–891.