Constructing Good Coefficient Functionals for Bivariate $C^1$ Quadratic Spline Quasi-Interpolants

27 July 2021 Original Citation:

Constructing Good Coefficient Functionals for Bivariate $C^1$ Quadratic Spline Quasi-Interpolants

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Sara Remogna. Constructing Good Coefficient Functionals for Bivariate

C

Quadratic Spline Quasi-Interpolants. Mathematical Methods for Curves

and Surfaces - Lecture Notes in Computer Science, 5862, 2010, DOI

10.1007/978-3-642-11620-9 22.

The definitive version is available at:

Bivariate

C

Quadratic Spline

Quasi-Interpolants

Sara Remogna

Universit`a degli Studi di Torino, Dipartimento di Matematica Via Carlo Alberto 10, 10123 Torino, Italy

sara.remogna@unito.it

Abstract. We consider discrete quasi-interpolants based on C1

quadra-tic box-splines on uniform criss-cross triangulations of a rectangular do-main. The main problem consists in finding good (if not best) coefficient functionals, associated with boundary box-splines, giving both an op-timal approximation order and a small infinity norm of the operator. Moreover, we want that these functionals only involve data points inside the domain. They are obtained either by minimizing an upper bound of their infinity norm w.r.t. a finite number of free parameters, or by inducing superconvergence of the operator at some specific points lying near or on the boundary.

Key words: Box-splines; Near-best quasi-interpolants; Superconvergence; Near-best linear functionals

1

Introduction

Let Ω = [0, hm]×[0, hn] be a rectangular domain divided into mn equal squares, m, n≤ 4, each of them being subdivided into 4 triangles by its diagonals. We denote by S1

2(Ω, T ) the space of C1 quadratic splines on the triangulation T of Ω obtained in this way. This space is generated by the (m + 2)(n + 2) spline functions {Bα, α ∈ A}, where A = {(i, j), 0 ≤ i ≤ m + 1, 0 ≤ j ≤ n + 1}, obtained by dilation/translation of the Zwart-Powell quadratic box-spline (ZP-element) ([5] Chap.1, [6] Chap.3).

The ZP-element is the bivariate C1 _{quadratic box-spline supported on the} octagon with center at the origin and vertices at (3₂,1₂), (1₂,3₂), (−1₂,3₂), (−3₂,1₂), (−3₂,−1₂), (−1₂,−3₂), (1₂,−3₂), (3₂,−1₂). It is strictly positive inside its support, that is partitioned into 28 triangular cells. On every cell the ZP-element is a polynomial of total degree 2 and in [7] the polynomials are given. The ZP-element can be also expressed in BB-form, i.e. specifying the Bernstein-B´ezier (abbr. BB)-coefficients on every triangular cell, [15].

In the space S1

2(Ω, T ) we consider discrete quasi-interpolants (abbr. dQI) of type

Qf =X

α∈A

where {λα, α∈ A} is a family of linear functionals which are local, in the sense that they are linear combinations of values of f at some points lying inside Ω and in a neighbourhood of the support Σα of Bα and such that Q is exact on the space P2 of quadratic polynomials.

The points Mα, α ∈ A, used in evaluating f are the centers of the squares (mn points), the midpoints of boundary segments (2(m + n) points) and the four vertices of Ω, see Fig.1 (see also Fig.1 in [12]). These points are defined by Mα= Mi,j= (si, tj) where

s0= 0, si= (i −1₂)h, 1 ≤ i ≤ m, sm+1= hm t0= 0, tj = (j −1₂)h, 1 ≤ j ≤ n, tn+1= hn

The values of the function f at those points are denoted by fα= f (Mα).

Fig. 1.Uniform criss-cross triangulation and data points.

Another spanning set for the space S1

2(Ω, T ) is formed by the box-splines {B′

α, α∈ A} constructed in [9] and [17], but, in some applications, it may be more convenient to make use of classical box-splines instead of the other ones, because the latter have different supports and different expressions in the domain, while the ZP-element is always the same.

The quasi-interpolants constructed in this paper are suitable for some appli-cations. For example they can be used for the approximation of the gradient of a function and the computation of critical points and curvatures of a surface [12]. Furthermore they can be used in the second stage of the two-stage method of Schumaker [19].

Here is an outline of the paper. In Section 2 we compare different choices for the functionals of such dQIs: near-best functionals and functionals inducing superconvergence at specific points. In Section 3 we estimate the operator norm by estimating the Lebesgue function near the origin of Ω for two particular dQIs. Finally, in Section 4, we give some numerical and graphical results.

2

On the Construction of Coefficient Functionals

In this section we are interested in the construction of linear functionals λα(f ) =

X β∈Fα

σα(β)f (Mβ), (2)

where the finite set of points {Mβ, β∈ Fα}, Fα⊂ A, lies in some neighbourhood of Σα∩ Ω and such that Qf ≡ f for all f in P2.

The construction of such coefficient functionals is related with the following differential QI (abbr. DQI), exact on P2 ([15] Chap. 6):

b Qf =X α  fα− h2 8 ∆fα  Bα. (3)

By using the five point discretisation of the Laplacian ∆, from (3), the fol-lowing discrete functionals λi,j are defined [16]

λi,j(f ) = 3 2fi,j−

1

8(fi−1,j+ fi+1,j+ fi,j−1+ fi,j+1). (4)

In the interior of the domain our quasi-interpolants make use of the same inner functionals λi,j, with i = 2, . . . , m − 1, j = 2, . . . , n − 1, defined by (4).

We propose two different ways of constructing functionals associated with the box splines whose supports are not entirely inside Ω: near-best functionals (de-noted by λ′

αand λ′′α) and functionals (denoted by λα) inducing superconvergence at some specific points.

2.1 Near-Best Boundary Functionals

In this section we construct convenient boundary coefficient functionals giving both the best approximation order and a small infinity norm of the operator. We call them near-best functionals and they are obtained by minimizing upper bounds for their infinity norm.

The method used in this subsection is closely related to the techniques given in [1, 2, 4, 14] to define near-best discrete quasi-interpolants on type-1 and type-2 triangulations (see also [3]).

From (2) it is clear that, for kf k∞≤ 1 and α ∈ A, |λα(f )| ≤ kσαk1 where σαis the vector with components σα(β), from which we deduce immediately

|Qf | ≤ X

α∈A

|λα(f )| Bα≤ max

α∈A|λα(f )| ≤ maxα∈Akσαk1,

therefore we can conclude

kQk∞≤ max

Now assuming that card(Fα) > 6, we can try to find σα∗ ∈ Rcard(Fα)solution of the minimization problem (see e.g. [4], [14] Chap.3)

kσ∗ αk1= min n kσαk1; σα∈ Rcard(Fα), Vασα= bα o ,

where Vασα= bαis a linear system expressing that Q is exact on P2. In our case we require that the coefficient functional coincides with the differential coefficient (f − 1

8∆f) at the center cα of the octagonal support of the box-spline Bα, for f ∈ P2.

This problem is a l1-minimization problem and there are many well-known techniques for approximating the solutions, not unique in general (cf. [20] Chap.6).

Hereinafter we analyse some coefficient functionals λ′

α near the boundary of the domain Ω.

Case α = (0, 0)

We consider the 10-point linear functional

λ′0,0(f ) = a1f0,0+a2(f1,0+f0,1)+a3(f2,0+f0,2)+a4(f3,0+f0,3)+a5f1,1+a6(f2,1+f1,2), and we impose λ′0,0(f ) ≡ (f − 18∆f)(c00), c00 = (−

1 2,−

1

2), for f ≡ 1, x, x2, xy. Due to symmetry of the ZP-element, there are only 6 unknowns and the mono-mials y and y2 can be excluded.

This leads to the system:

a1+ 2a2+ 2a3+ 2a4+ a5+ 2a6= 1, a2+ 3a3+ 5a4+ a5+ 4a6= −1, a2+ 9a3+ 25a4+ a5+ 10a6= 0, a5+ 6a6= 1,

whose solution depends on the two parameters a4and a6 a1= 14 3 − 16 3 a4− 4a6, a2= − 5 2+ 5a4+ 5a6, a3= 1 6− 10 3 a4− a6, a5= 1 − 6a6. If we minimize the norm kλ′0,0k∞we obtain

a1= 20 9 , a2= 0, a3= − 10 9 , a4= 1 3, a5= 0, a6= 1 6,

with a norm equal to kλ′0,0k∞ = 5.44. Graphically we obtain a functional with data points shown in Fig.2(a).

If we want a functional with a smaller norm, we can add e.g. the point M22 λ′′0,0(f ) = a1f0,0+ a2(f1,0+ f0,1) + a3(f2,0+ f0,2) + a4(f3,0+ f0,3) + a5f1,1+

+a6(f2,1+ f1,2) + a7f2,2.

Solving the corresponding system and minimizing the norm kλ′′0,0k∞, we obtain a1= 22 9 , a2= 0, a3= − 1 9, a4= 0, a5= − 3 2, a6= 0, a7= 5 18, with a norm equal to kλ′′0,0k∞= 4.44. Graphically we obtain the functional with data points shown in Fig.2(b).

(a) (b) Fig. 2.Near-best functionals associated with B0,0.

Case α = (1, 0)

We consider λ′1,0(f ) = a1f0,0+ a2f1,0+ a3f0,1+ a4f2,0+ a5f0,2+ a6f3,0+ a7f0,3+ a8f1,1+ a9f2,1+ a10f1,2. Solving the corresponding system and minimizing the norm we obtain

a1= 3845, a2= a3= 0, a4=1918, a5= −12, a6= −1130, a7= 103, a8= a9= 0, a10= −13,

with a norm equal to kλ′1,0k∞= 3.4, see Fig.3.

Case α = (2, 0)

We consider λ′2,0(f ) = a1f2,0+ a2(f1,0+ f3,0) + a3f2,1+ a4(f1,1+ f3,1) + a5f2,2+ a6(f1,2+ f3,2). Solving the corresponding system and minimizing the norm we obtain a1= 7 3, a2= 0, a3= − 13 12, a4= − 5 24, a5= 0, a6= 1 12, with a norm equal to kλ′2,0k∞= 4, see Fig.4(a).

If we add the point M23, λ

′′ 2,0(f ) = a1f2,0+ a2(f1,0+ f3,0) + a3f2,1+ a4(f1,1+ f3,1) + a5f2,2+ a6(f1,2+ f3,2) + a7f2,3, we get a1= 23 15, a2= a3= a4= 0, a5= − 7 12, a6= − 1 8, a7= 3 10, with a norm equal to kλ′′2,0k∞= 2.67, see Fig.4(b).

Case α = (1, 1) We consider λ′

1,1(f ) = a1f1,1+ a2(f2,1+ f1,2) + a3(f3,1+ f1,3) + a4f2,2. Solving the linear system we obtain

a1= 3 4, a2= 1 4, a3= − 1 8, a4= 0, with a norm equal to kλ′

1,1k∞= 1.5, see Fig.5(a).

Case α = (2, 1)

We consider λ′2,1(f ) = a1f2,1+ a2(f1,1+ f3,1) + a3f2,2+ a4(f1,2+ f3,2) + a5f2,3+ a6(f1,3+ f3,3). Solving the corresponding system and minimizing the norm we obtain a1= 7 8, a2= 0, a3= 3 8, a4= − 1 16, a5= 0, a6= − 1 16, with a norm equal to kλ′

2,1k∞= 1.5, see Fig.5(b).

Fig. 3.Near-best functional associated with B1,0.

2.2 Boundary Functionals Inducing Superconvergence

In this section we construct boundary coefficient functionals inducing supercon-vergence of the operator Q at some specific points, the vertices Ak,l= (kh, lh), k = 0, . . . , m, l = 0, . . . , n and the centers Mk,l, k = 1, . . . , m, l = 1, . . . , n of squares, the midpoints Ck,l of horizontal edges Ak−1,lAk,l, k = 1, . . . , m, l= 0, . . . , n and the midpoints Dk,l, k = 0, . . . , m, l = 1, . . . , n of vertical edges

(a) (b) Fig. 4.Near-best functionals associated with B2,0.

(a) (b)

Fig. 5.Near-best functionals associated with B1,1and B2,1.

Ak,l−1Ak,l, see Fig.6. We remark that if A = Z2the discrete operator defined by using the coefficient functionals given in (4) is superconvergent at these points [11, 18].

Recall that we construct boundary coefficient functionals λα(f ) so that they coincide with the differential coefficients (f − 1

8∆f) at the center cα of the octagonal support of the box-splines Bα for f ∈ P2.

Since the differential quasi-interpolant (3) is exact on P2, the discrete oper-ator that we are constructing is also exact on P2, therefore the approximation order f − Qf is O(h3_{) for smooth functions.}

If we want superconvergence at some specific points, i.e. f (M ) − Qf (M ) = O(h4_{), we have to require that, for f ∈ P}

3, the quasi-interpolant Q interpolates the function f at those points. So we impose that Qf (M ) = f (M ) for f ∈ P3\P2, M being a specific point of the domain.

This leads to a system of equations. We consider systems with free parameters and we choose them by minimizing an upper bound of the infinity norm kλαk∞ and solving the corresponding l1-minimization problem. Hereinafter we analyse some coefficient functionals λα near the boundary of Ω.

Case α = (2, 1)

We consider 12 points and 8 unknowns

λ2,1(f ) = a1f2,1+ a2(f1,1+ f3,1) + a3f2,2+ a4(f1,2+ f3,2) + a5f2,3+ +a6(f1,3+ f3,3) + a7f2,0+ a8(f1,0+ f3,0).

We require that:

(i) λ2,1(f ) coincides with the differential coefficients (f − 1₈∆f)(c2,1), c2,1 = (3

2, 1

2) for f ∈ P2, i.e. for f ≡ 1, x, y, x

2_{, xy, y}2_;

(ii) Qf (M ) = f (M ) for f ∈ P3\P2, i.e. for f ≡ x3, x2y, xy2, y3, and M = (2, 1), (3 2, 3 2), (2, 3 2).

This leads to a system whose solution depends on the two parameters a4and a6

a1= 15₈ + 6a4+ 10a6, a2= −₈1− 3a4− 5a6, a3= −1₄− 2a4, a5= ₄₀1 − 2a6, a7= −₅2− 8a6− 4a4, a8= 2a4+ 4a6. If we minimize the norm kλ2,1k∞we obtain

a1= 13 8 , a2= 0, a3= − 1 6, a4= − 1 24, a5= 1 40, a6= 0, a7= − 7 30, a8= − 1 12, with a norm equal to kλ2,1k∞ = 2.3. Graphically we obtain a functional with data points shown in Fig.7.

Fig. 7.Functional inducing superconvergence associated with B2,1.

Case α = (1, 1)

We consider λ1,1(f ) = a1f0,0+ a2(f1,0+ f0,1) + a3f1,1+ a4(f2,0+ f0,2) + a5(f2,1+ f1,2) + a6(f3,0+ f0,3) + a7f2,2+ a8(f3,1+ f1,3).

We require that:

(i) λ1,1(f ) coincides with the differential coefficients (f −1₈∆f)(1₂,1₂) for f ∈ P2; (i) Qf (M ) = f (M ) for f ∈ P3\P2 and M = (1, 1), (3₂,1).

Solving the corresponding system and minimizing the norm we obtain a1= − 4 15, a2= 0, a3= 33 20, a4= − 2 15, a5= − 1 20, a6= 0, a7= − 1 15, a8= 1 40, with a norm equal to kλ1,1k∞= 2.4, see Fig.8(a).

Case α = (2, 0)

We consider λ2,0(f ) = a1f2,1+ a2(f1,1+ f3,1) + a3f2,2+ a4(f1,2+ f3,2) + a5f2,3+ a6(f1,3+ f3,3) + a7f2,0+ a8(f1,0+ f3,0).

We require that:

(i) λ2,0(f ) coincides with the differential coefficients (f −1₈∆f)(3₂,−1₂) for f ∈ P2;

(i) Qf (M ) = f (M ) for f ∈ P3\P2 and M = (2, 0), (32, 1 2), (2,

1 2). Solving the corresponding system and minimizing the norm we obtain

a1= − 9 8, a2= − 1 4, a3= 0, a4= 1 8, a5= − 1 40, a6= 0, a7= 12 5 , a8= 0, with a norm equal to kλ2,0k∞= 4.3, see Fig.8(b).

Case α = (1, 0)

We consider λ1,0(f ) = a1f0,0+ a2f1,0+ a3f2,0+ a4f3,0+ a5f0,1+ a6f1,1+ a7f2,1+ a8f0,2+ a9f1,2+ a10f0,3+ a11f4,0+ a12f3,1+ a13f2,2+ a14f1,3.

We require that:

(i) λ1,0(f ) coincides with the differential coefficients (f −1₈∆f)(1₂,−1₂) for f ∈ P2;

(i) Qf (M ) = f (M ) for f ∈ P3\P2 and M = (1, 0), (1₂,1₂), (1,1₂). Solving the corresponding system and minimizing the norm we obtain

a1= 0, a2=131₆₀, a3=₄₀9, a4= 0, a5= −173₃₀₀, a6= −13₄₀, a7= −47₆₀, a8= a9= 0, a10= ₂₀3, a11= −₁₂₀1 , a12=₅₀3, a13=1₄, a14= −₄₀7, with a norm equal to kλ1,0k∞= 4.74, see Fig.9(a).

Case α = (0, 0)

We consider λ0,0(f ) = a1f0,0+ a2(f1,0+ f0,1) + a3f1,1+ a4(f2,0+ f0,2) + a5(f2,1+ f1,2) + a6(f3,0+ f0,3) + a7f2,2+ a8(f3,1+ f1,3) + a9(f4,0+ f0,4) + a10(f4,1+ f1,4) + a11(f2,3+ f3,2).

We require that:

(i) λ0,0(f ) coincides with the differential coefficients (f − 1₈∆f)(−1₂,−1₂) for f ∈ P2;

(i) Qf (M ) = f (M ) for f ∈ P3\P2 and M = (0, 0), (12,0).

Solving the corresponding system and minimizing the norm we obtain a1= 1403₅₀₄, a2= 0, a3= −63₃₂, a4= −₁₄₄₀397, a5= a6= 0, a7= 317₂₈₈,

a8= 0, a9= ₂₂₄11, a10= 0, a11= −₁₆₀37, with a norm equal to kλ0,0k∞= 7.0, see Fig.9(b).

3

Two Examples of Discrete Quasi-Interpolants

Now, choosing conveniently the boundary coefficient functionals, we define two different discrete QIs of type (1) and we study their infinity norm on some square cells of the domain Ω, as shown in Fig.10. We recall that in the interior of the domain the coefficient functionals λi,j, with i = 2, . . . , m − 1, j = 2, . . . , n − 1 are defined in (4).

(a) _(b)

Fig. 8.Functionals inducing superconvergence associated with B1,1and B2,0.

(a) (b)

Fig. 10.Some square cells near the origin.

3.1 The Discrete Quasi-Interpolant Q1f A first discrete QI is defined by

Q1f = X α∈A

λα(f )Bα,

where the boundary functionals are near-best in the sense of Section 2.1. Thanks to symmetry properties, only functionals in the neighbourhood of the origin are required:

λ0,0(f ) = 229f0,0−19(f2,0+ f0,2) −32f1,1+185f2,2,

λ1,0(f ) = 3845f0,0+1918f2,0−12f0,2−1130f3,0+103f0,3−13f1,2, λ1,1(f ) = 34f1,1+14(f2,1+ f1,2) −18(f3,1+ f1,3).

Along the lower edge, for i = 2, . . . , m − 1, we have

λi,0(f ) = 23₁₅fi,0−₁₂7fi,2−1₈(fi−1,2+ fi+1,2) +₁₀3fi,3,

λi,1(f ) = 7₈fi,1+3₈fi,2−₁₆1(fi−1,2+ fi+1,2) −₁₆1(fi−1,3+ fi+1,3), and analogous formulae along the three other edges of Ω.

3.2 The Discrete Quasi-Interpolant Q2f A second discrete QI is defined by

Q2f = X α∈A

µα(f )Bα,

where we choose boundary functionals inducing superconvergence at specific points, defined in Section 2.2. Thanks to symmetry properties, only functionals

in the neighbourhood of the origin are required: µ0,0(f ) = 1403₅₀₄f0,0−₁₄₄₀397(f2,0+ f0,2) −63₃₂f1,1+₂₂₄11(f4,0+ f0,4)+ +317₂₈₈f2,2−₁₆₀37(f3,2+ f2,3), µ1,0(f ) = 131₆₀f1,0+₄₀9f2,0−₁₂₀1 f4,0−173₃₀₀f0,1−13₄₀f1,1−47₆₀f2,1+ +₅₀3f3,1+1₄f2,2+₂₀3f0,3−₄₀7f1,3, µ1,1(f ) = −₁₅4f0,0−₁₅2(f2,0+ f0,2) +33₂₀f1,1−₂₀1(f2,1+ f1,2)− −₁₅1f2,2+₄₀1(f3,1+ f1,3).

Along the lower edge, for i = 2, . . . , m − 1, we have µi,0(f ) = 12₅fi,0−9₈fi,1−1₄(fi−1,1+ fi+1,1)+

+1₈(fi−1,2+ fi+1,2) −₄₀1fi,3,

µi,1(f ) = −₃₀7fi,0−₁₂1(fi−1,0+ fi+1,0) +13₈fi,1−1₆fi,2− −₂₄1(fi−1,2+ fi+1,2) +₄₀1fi,3,

and analogous formulae along the three other edges of Ω.

3.3 Norm Estimates

Now, we study the infinity norms of both operators, Q1 and Q2.

We define the quasi-Lagrange functions L(v)α , v = 1, 2, by the following ex-pression of Q1f and Q2f Q1f = X α∈A λα(f )Bα= X α∈A fαL(1)α , Q2f = X α∈A µα(f )Bα= X α∈A fαL(2)α .

We know that the infinity norm of such operators is equal to the Chebyshev norm of its Lebesgue function defined by:

Λv= X α∈A

|L(v)α |, v = 1, 2.

Using a computer algebra system, we have computed the maxima of Λv in some square cells near the origin (south-west vertex of Ω), see Fig.10, for both operators. The approximate values of these maxima are:

Table 1.Maximum value of the Lebesgue functions Λv, v = 1, 2 near the origin.

(i, j) (0,0) (1,0) (2,0) (1,1) (2,1) (2,2) Λ1|Ωi,j 2.00 1.96 1.65 1.63 1.50 1.50

Theorem 1. For the operator Qv, v = 1, 2 the following bounds are valid kQvk∞≤ v, v = 1, 2.

We remark that if A = Z2 _{the two discrete operators Q}

1 and Q2 coincide and their norm is bounded by 1.5 [12].

4

Numerical Results

In this section we present some numerical results obtained by a computational procedure developed in a Matlab environment [8].

We approximate the following functions (Figs.11÷13) f1(x, y) =

1

1 + x2_{+ y}2, f2(x, y) = ln(1 + x 2_{+ y}2_),

on the square [−1, 1] × [−1, 1], and the Franke’s function (see e.g. [13]) f3(x, y) = 3₄exp −1₄ (9x − 2)2+ (9y − 2)2

+ +3 4exp  −(9x+1)₄₉ 2 −(9y+1)₁₀ 2+ +1₂exp −1₄ (9x − 7)2_{+ (9y − 3)}2

₋ −1₅exp − (9x − 4)2_{+ (9y − 7)}2

on the square [0, 1] × [0, 1]. Fig. 11.Function f1(x, y) on [−1, 1] × [−1, 1].

For each test function, we compute, using a 2000 × 2000 uniform rectangular grid of evaluation points in the domain, the maximum absolute error Evf = kf − Qvfk∞, v = 1, 2, for increasing values of m and n, and the logarithm of the ratio between two consecutive errors, rvf, v = 1, 2, see Table 2. In Figs.14÷16

Fig. 12.Function f2(x, y) on [−1, 1] × [−1, 1]. Fig. 13.Function f3(x, y) on [0, 1] × [0, 1]. (a) (b) Fig. 14.|Q1f1− f1| ≤ 6.7495 · 10− 6 and |Q2f1− f1| ≤ 1.9089 · 10− 6 with m = n = 64.

(a) (b) Fig. 15.|Q1f2− f2| ≤ 7.5843 · 10− 6 and |Q2f2− f2| ≤ 1.1194 · 10− 6 with m = n = 64. (a) (b) Fig. 16.|Q1f3− f3| ≤ 1.0417 · 10− 4 and |Q2f3− f3| ≤ 5.0880 · 10− 5 with m = n = 64.

Table 2.Maximum absolute errors. m= n E1f1 r1f1 E2f1 r2f1 32 5.5137(-005) 2.1396(-005) 64 6.7495(-006) 3.03 1.9089(-006) 3.49 128 8.4039(-007) 3.01 2.0738(-007) 3.20 256 1.0472(-007) 3.00 2.4737(-008) 3.07 512 1.3068(-008) 3.00 3.0335(-009) 3.03 1024 1.6320(-009) 3.00 3.7690(-010) 3.01 m= n E1f2 r1f2 E2f2 r2f2 32 6.3773(-005) 1.1627(-005) 64 7.5843(-006) 3.07 1.1194(-006) 3.38 128 9.2326(-007) 3.04 1.2723(-007) 3.14 256 1.1313(-007) 3.03 1.5380(-008) 3.05 512 1.3957(-008) 3.02 1.8994(-009) 3.02 1024 1.7389(-009) 3.00 2.3611(-010) 3.01 m= n E1f3 r1f3 E2f3 r2f3 32 9.1190(-004) 7.2117(-004) 64 1.0417(-004) 3.13 5.0880(-005) 3.83 128 1.2172(-005) 3.10 4.1378(-006) 3.62 256 1.4577(-006) 3.06 4.1153(-007) 3.33 512 1.7719(-007) 3.04 4.7279(-008) 3.12 1024 2.1932(-008) 3.01 5.7982(-009) 3.03

the corresponding errors between f and the two quasi-interpolants Q1f and Q2f with m = n = 64 are shown.

If we evaluate the error at the points where superconvergence holds, Fig.6, we can observe that with the operator Q2 the error is O(h4), see Table 3.

Although the infinity norm of the operator Q2 is greater than the infinity norm of Q1, we can notice that the overall error is smaller.

Furthermore, from Figs.14(b)÷16(b), it seems that the operator Q2has a good global error while the operator Q1 presents a greater error near the boundary as shown in Figs.14(a)÷16(a).

5

Final Remarks

In this paper we have defined and analysed C1 _{quadratic discrete} quasi-interpo-lants, constructing their coefficient functionals in several ways, comparing them and giving norm estimates.

In a similar way, we are investigating the construction of good coefficient functionals for C2_{quartic spline quasi-interpolants on criss-cross triangulations} and C2 _{cubic spline quasi-interpolants on Powell-Sabin triangulations [10].}

We plan to use these quasi-interpolants in the second stage of the two-stage method of Schumaker [19].

Furthermore we will apply these operators to the approximation of critical points and curvatures of a surface [12].

Table 3.Maximum absolute errors at specific points (Section 2.2). m= n E1f1 r1f1 E2f1 r2f1 32 5.3905(-005) 1.8852(-005) 64 6.7482(-006) 3.00 1.1886(-006) 3.99 128 8.4015(-007) 3.01 7.4451(-008) 4.00 256 1.0470(-007) 3.00 4.6558(-009) 4.00 512 1.3064(-008) 3.00 2.9102(-010) 4.00 1024 1.6314(-009) 3.00 1.8190(-011) 4.00 m= n E1f2 r1f2 E2f2 r2f2 32 6.2292(-005) 9.4628(-006) 64 7.4389(-006) 3.07 5.9488(-007) 3.99 128 9.0720(-007) 3.04 3.7235(-008) 4.00 256 1.1196(-007) 3.02 2.3280(-009) 4.00 512 1.3905(-008) 3.01 1.4551(-010) 4.00 1024 1.7325(-009) 3.00 9.0949(-012) 4.00 m= n E1f3 r1f3 E2f3 r2f3 32 8.7514(-004) 7.1035(-004) 64 1.0120(-004) 3.11 4.7068(-005) 3.92 128 1.1913(-005) 3.09 3.0205(-006) 3.96 256 1.4376(-006) 3.05 1.9001(-007) 3.99 512 1.7638(-007) 3.03 1.1886(-008) 4.00 1024 2.1836(-008) 3.01 7.4321(-010) 4.00

Acknowledgement: The author is grateful to Prof. C. Dagnino and Prof. P. Sablonni`ere for helpful discussions and comments.

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