2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 157–166. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

APPROXIMATION OF DISCONTINUOUS CURVES AND SURFACES BY DISCRETE SPLINES WITH TANGENT CONDITIONS

ABDELOUAHED KOUIBIA, MIGUEL PASADAS

Abstract. This paper concerns the construction of a discontinuous parametric curve or surface from a finite set of points and tangent conditions. The method is adapted from the theory of the discrete smoothing variational splines to introduce a discontinuity set and some tangent conditions. Such method is justified by a convergence result.

1. Introduction

The problem of the construction of discontinuous parametric curves and surfaces from some Lagrangean points and a set of tangent spaces is frequently encountered in CAGD, Geology, and other Earth Sciences.

The authors in [10] present a smoothing method for fitting parametric surfaces from sets of data points and tangent planes. In those papers and in [6, 7, 8, 10], the corresponding original curves or/and surfaces that are approximated do not present any discontinuities. So for its practical interest, we introduce some discontinuity in order to study an approximation problem of discontinuous curves and surfaces.

In this paper we present a discrete version of the previous problem in a finite element space by minimizing a quadratic functional from a set of Lagrangean data and another one of tangent conditions. The approximation of discontinuous functions from a set of scattered data points is usually a two steps: first, a detection algorithm is applied to localize the discontinuity sets, then the functions are reconstructed using a fitting method. In this work we propose a method for the second stage, based on the computation of discrete smoothing variational splines [7].

Specially, we study the following problem: given a differentiable function f in a subset Ω′ of an open set Ω ⊂ ℝ^{p} with values in ℝ^{n}, 1 ≤ p < n ≤ 3, whose first partial derivatives or she can present discontinuity in a subset F of \overline{Ω}, such that Ω′ = Ω\\overline{F}, construct a function σ that approximates f in the given Lagrangean points of Ω′ and whose tangent spaces at the points of an other given set of Ω′ are close to the tangent spaces of f at the same points. To do this, firstly using the work of Arcangéli, Manzanilla and Torrens [1], we determine certain hypotheses about the set Ω′ that allows to model the contingent discontinuities of f. Secondly, we study a method of smoothness which results from adapting to this context the theory of the discrete smoothing variational splines (c.f. [7]).

2000 Mathematics Subject Classification. 65D07, 65D10, 65D17.

Key words and phrases. Discrete smoothing; curve; surface; discontinuity; spline;

tangent condition.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

In Section 2, we briefly recall preliminary notation and some results. We study the problem of discrete smoothing variational splines with tangent conditions in Section 3. Section 4 is devoted to compute such spline and to prove a convergence result.

2. Notation

We denote by ⟨·⟩_{ℝ^{n}} and ⟨ · , · ⟩_{ℝ^{n}} respectively, the Euclidean norm and the inner product in ℝ^{n}, with n, m and p belonging to ℕ^{∗}. We denote by \overline{E}, δE and cardE respectively, the adherence, the bounded and the cardinal of E for each subset E of ℝ^{p}. Let us consider ℝ^{N,n} the space of real matrices with N rows and n columns equipped with the inner product

⟨A, B⟩_{N,n} = ∑_{i=1}^{N} ∑_{j=1}^{n} a_{ij}b_{ij}

and the corresponding norm

⟨A⟩_{N,n} = ⟨A, A⟩_{N,n}^{1/2}.

For all α = (α_{1}, …, α_{p}) ∈ ℕ^{p}, we write |α| = ∑_{i=1}^{p} α_{i} and we indicate by ∂^{α} the operator of partial derivative

∂^{α} = \frac{∂^{|α|}}{∂x_{1}^{α_{1}} … ∂x_{p}^{α_{p}}}

Let ω be a nonempty open bounded set of ℝ^{p} and we denote by H^{m}(ω; ℝ^{n}) the usual Sobolev space of (classes of) functions u belonging to L^{2}(ω; ℝ^{n}), together with all their partial derivative ∂^{α}u –in the distribution sense– of order |α| ≤ m. This space is equipped with the inner product of order ℓ

( u , v)_{ℓ,ω,ℝ^{n}} = ( ∑_{|α|=ℓ} ∫_{ω}⟨∂^{α}u(x), ∂^{α}v(x)⟩_{ℝ^{n}})^{1/2}, ℓ = 0, …, m,

the corresponding semi–norms of order ℓ

| u |_{ℓ,ω,ℝ^{n}} = ( u , u)_{ℓ,ω,ℝ^{n}}^{1/2}, ℓ = 0, …, m,

the norm

∥ u ∥_{m,ω,ℝ^{n}} = ( ∑_{ℓ=0}^{m} | u |_{ℓ,ω,ℝ^{n}}^{2})^{1/2}

and the corresponding inner product

(( u , v))_{m,ω,ℝ^{n}} = ∑_{ℓ=0}^{m}( u , v)_{ℓ,ω,ℝ^{n}}.

Given a function f : ω → ℝ^{n}, we denote by ImDf(x) the image of the differential of f at the point x ∈ ω, when this exists, i. e. the linear subspace generated by {∂^{α}f(x) : |α| = 1}. Furthermore, if 1 ≤ p < n ≤ 3, we can consider f as the parameterization of a curve (p = 1) or a surface (p = 2) and, if f is differentiable at x ∈ ω, the space ImDf(x) is called the tangent space of f at x, sometimes when p = 2 it is written by T_{x}(f) = span⟨D_{1}f(x), D_{2}f(x)⟩, where D_{1}f and D_{2}f denote the first partial derivatives of f.

3. Discrete variational spline over Ω′ with tangent conditions

The first step in order to develop this work is to have an adequate characterization over a set of discontinuity. Let us introduce the following definition due to R. Arcangéli, R. Manzanilla and J. J. Torrens [1].

Definition 3.1. Let Ω be a bounded open connected set of ℝ^{p} with Lipschitz boundary and let F be a nonempty subset of \overline{Ω} such that, there exists a finite family {R_{1}, …, R_{I}} of open connected subsets of Ω with Lipschitz boundary, verifying the following conditions:

(i) for all i, j = 1, …, I, i ≠ j, R_{i} ∩ R_{j} = ∅; (ii) ⋃_{i=1}^{I} \overline{R}_{i} = \overline{Ω}; (iii) F ⊂ δR, where ∪_{i=1}^{I}R_{i} = R; (iv) F is contained in the interior of δΩ (equipped of the induced topology by

ℝ^{p}) of F ∩ δΩ;

v) the interior in δR of \overline{F} ∩ Ω is contained in F; vi) \overline{F} ∩ δΩ is contained in F.

It is said that the family \{R_{1}, …, R_{I}\} represents F in Ω and we write Ω′ = Ω\\overline{F}.

We denote by C_{F}^{k} (Ω′; ℝ^{n}) the space of functions φ ∈ C^{k}(Ω′; ℝ^{n}) such that

∀i = 1, …, I, φ|_{R_{i}} ∈ C^{k}(\overline{R}_{i}; ℝ^{n}).

Such space is equipped by the norm

∥φ∥_{C_{F}^{k} (Ω′;ℝ^{n})} = max_{1≤i≤I} ∥φ|_{R_{i}}∥_{C^{k}(\overline{R}_{i};ℝ^{n})} .  (3.1)

Now, we suppose that

m > \frac{p}{2} + 1.  (3.2)

Let Υ_{0} be a curve or surface parameterized by a function f ∈ H^{m}(Ω′; ℝ^{n}), and A_{1}, A_{2} be two ordered finite subsets of, respectively, N_{1} and N_{2} distinct points of \overline{Ω}. For any a ∈ A_{1}, let us consider the linear form defined on C_{F}^{0} (Ω′; ℝ^{n}) by

ϕ_{a}v = \begin{cases}v(a) & if a ∈ A_{1}\F, \\ v|_{R_{i}}(a) & if a ∈ A_{1} ∩ R_{i} ∩ F, 1 ≤ i ≤ I, \end{cases}  (3.3)

and, for any a ∈ A_{2} let Π_{a} be the operator defined on C_{F}^{1} (Ω′; ℝ^{n}) by

Π_{a}v = \begin{cases}(P_{S_{a}^{⊥}}(\frac{∂v}{∂x_{j}} (a)))_{1≤j≤p} & if a ∈ A_{2}\F, 1 ≤ i ≤ I, \\ (P_{S_{a}^{⊥}}(^{\frac{∂v|_{R_{i}}}{∂x_{j}}} (a)))_{1≤j≤p} & if a ∈ A_{2} ∩ R_{i} ∩ F, 1 ≤ i ≤ I, \end{cases}  (3.4)

where, for any a ∈ A_{2}, P_{S_{a}^{⊥}} is the operator projection onto S_{a}^{⊥}, being S_{a}^{⊥} the orthogonal complement of the linear space S_{a} = ImDf(a). Finally, let

Lv = (ϕ_{a}v)_{a∈A_{1}} and Πv = (Π_{a}v)_{a∈A_{2}}

and we suppose that

KerL ∩ P̃_{m−1}(Ω′; ℝ^{n}) = {0}  (3.5)

where P̃_{m−1}(Ω′; ℝ^{n}) designs the space of functions over Ω′ into ℝ^{n} that are polynomials of total degree ≤ m − 1 respect to the set of variables over each connected component of Ω′.

Now, suppose we are given:

• a subset H of (0, +∞) with 0 is an accumulation point;

• for all h ∈ H, a partition T_{h} of \overline{Ω} made with rectangles or triangles K of disjoint interiors and diameter h_{K} ≤ h such that

∀ K ∈ T_{h}, K^{∘} ∩ F = ∅;  (3.6)
each side of K is a side of another K′ or a part of δΩ or a part of F; (3.7)

• for any h ∈ H, a finite element space X_{h} constructed on T_{h} such that

X_{h} ⊂ H^{m}(Ω′) ∩ C_{F}^{k} (Ω′), k + 1 ≥ m;  (3.8)

• for any h ∈ H, a parametric finite element space V_{h} constructed from X_{h} by V_{h} = (X_{h})^{n}, and from (3.8) satisfies

V_{h} ⊂ H^{m}(Ω′; ℝ^{n}) ∩ C_{F}^{k} (\overline{Ω′}; ℝ^{n}).  (3.9)

Now, given τ ≥ 0 and ε > 0, let J_{ετ} be the functional defined on V_{h} by

J_{ετ} (v) = ⟨Lv − Lf⟩_{N_{1},n}^{2} + τ⟨Πv⟩_{N_{2},pn}^{2} + ε |v|_{m,Ω′,ℝ^{n}}^{2} .  (3.10)

Remark 3.2. We observe that the functional J_{ετ} (v) contains different terms which can be interpreted as follows:

• The first term, ⟨Lv − Lf⟩_{N_{1},n}^{2}, indicates how well v approaches f in a discrete least squares sense.

• The second term, ⟨Πv⟩_{N_{2},pn}^{2}, indicates how well, for any point a ∈ A_{2}, the tangent spaces ImDf(a) and ImDv(a) are really close.

• The last term, |v|_{m,Ω′,ℝ^{n}}^{2}, measures the degree of smoothness of v in order

to reduce, as much as possible, any unwanted oscillations. We note that the parameters τ and ε control the relative weights corresponding, respectively, to the last two terms.

Now, we consider the following minimization problem: Find an approximating curve or surface Υ of Υ_{0} parameterized by a function σ_{ετ}^{h} belonging to V_{h} from the data {f(a) : a ∈ A_{1}} and {S_{a} : a ∈ A_{2}}, such that σ_{ετ}^{h} minimizes the functional J_{ετ} on V_{h}, i.e. find σ_{ετ}^{h} such that

σ_{ετ}^{h} ∈ V_{h}, and for all v ∈ V_{h}, J_{ετ} (σ_{ετ}^{h}) ≤ J_{ετ} (v).  (3.11)

Theorem 3.3. The problem (3.11) has a unique solution, called discrete smoothing variational spline with tangent conditions in Ω′ relative to A_{1}, A_{2}, f, τ and ε, which is also the unique solution of the following variational problem: Find σ_{ετ}^{h} such that σ_{ετ} ∈ V_{h} and for all v ∈ V_{h},

⟨Lσ_{ετ}^{h} , Lv⟩_{N_{1},n} + τ⟨Πσ_{ετ}^{h} , Πv⟩_{N_{2},pn} + ε(σ_{ετ}^{h} , v)_{m,Ω′,ℝ^{n}} = ⟨Lf, Lv⟩_{N_{1},n}.

Proof. Taking into account (3.2), (3.5) and that the norm

[[v]] = (⟨Lv⟩_{N_{1},n}^{2} + τ⟨Πv⟩_{N_{2},pn}^{2} + ε|v|_{m,Ω′,ℝ^{n}}^{2})^{1/2}

is equivalent in V_{h} to the norm ∥ · ∥_{m,Ω′,ℝ^{n}} (cf. [1, Proposition 4.1]), one easily checks that the symmetric bilinear form ã : V_{h} × V_{h} ⟶ ℝ given by

ã(u, v) = ⟨Lu, Lv⟩_{N_{1},n} + τ⟨Πu, Πv⟩_{N_{2},pn} + ε(u, v)_{m,Ω′,ℝ^{n}}

is continuous and V_{h}–elliptic. Likewise, the linear form

φ : v ∈ V_{h} ⟼ φ(v) = ⟨Lf, Lv⟩_{N_{1},n}

is continuous. The result is then a consequence of the Lax–Milgram Lemma (see [3]). □

4. Computation and convergence result

Let us see how to compute the discrete smoothing variational spline with tangent conditions. To do this, for any h ∈ H, let I and {w_{1}, …, w_{I}} be the dimension and a basis of X_{h}, respectively, and let us denote by {e_{1}, e_{2}, …, e_{n}} the canonical basis of ℝ^{n}. Then, the family {v_{1}, …, v_{Z}}, with Z = nI, is a basis of V_{h}, where for i = 1, …, I and ℓ = 1, 2, …, n,

j = n(i − 1) + ℓ, v_{j} = w_{i}e_{ℓ}.

Thus, for any h ∈ H, the function σ_{ετ}^{h} can be expressed as

σ_{ετ}^{h} = ∑_{i=1}^{Z} β_{i}v_{i},

with the unknown β_{i} ∈ ℝ, for i = 1, …, Z.

Applying Theorem 3.3, we obtain that the vector β = (β_{i})_{1≤i≤Z} ∈ ℝ^{Z} is the solution of the following linear system of order Z:

(A^{T} A + τ P^{T} P + εR)β = A^{T} b,  (4.1)

where

A = (v_{j}(a_{i}))_{1≤i≤N_{1} 1≤j≤Z};
P = (Π_{a_{i}}v_{j})_{1≤i≤N_{2} 1≤j≤Z};
R = ((v_{i}, v_{j})_{m,Ω′,ℝ^{n}})_{1≤i,j≤Z};
b = (f(a_{i}))_{1≤i≤N_{1}}.

We point out that A^{T} A + τP^{T} P + εR of the linear system given in (4.1) is a band matrix which is symmetric positive definite.

Now, under adequate hypotheses, we shall show that the discrete smoothing variational spline with tangent conditions converges to f. Suppose that we are given:

• a subset D of (0, +∞) with 0 as an accumulation point;

• for all d ∈ D, two subsets A_{1}^{d} and A_{2}^{d} of respectively N_{1} = N_{1}(d) and N_{2} = N_{2}(d) distinct points of \overline{Ω};

• for all d ∈ D and any a ∈ A_{1}^{d}, let us consider the linear form defined on C_{F}^{0} (Ω′; ℝ^{n}) by

ϕ_{a}^{d}v = \begin{cases}v(a) & if a ∈ A_{1}^{d}\F, \\ v|_{R_{i}}(a) & if a ∈ A_{1}^{d} ∩ R_{i} ∩ F, 1 ≤ i ≤ I; \end{cases}

• for all d ∈ D and any a ∈ A_{2}^{d}, let Π_{a}^{d} be the operator defined in C_{F}^{1} (Ω′; ℝ^{n}) by

Π_{a}^{d}v = \begin{cases}(P_{S_{a}^{⊥}}(\frac{∂v}{∂x_{j}} (a)))_{1≤j≤p} & if a ∈ A_{2}^{d}\F, 1 ≤ i ≤ I, \\ (P_{S_{a}^{⊥}}(\frac{∂v|_{R_{i}}}{∂x_{j}} (a)))_{1≤j≤p} & if a ∈ A_{2}^{d} ∩ R_{i} ∩ F, 1 ≤ i ≤ I, \end{cases}

where for any a ∈ A_{2}^{d}, P_{S_{a}^{⊥}} is the operator projection onto S_{a}^{⊥}, being S_{a}^{⊥} the orthogonal complement of the linear space S_{a} = ImDf(a).

Finally, for any d ∈ D let

L^{d}v = (ϕ_{a}^{d}v)_{a∈A_{1}^{d}} and Π^{d}v = (Π_{a}^{d}v)_{a∈A_{2}^{d}} .

We suppose that

ker L^{d} ∩ P̃_{m−1}(Ω′; ℝ^{n}) = {0}  (4.2)

and that

sup_{x∈Ω′} min_{a∈A_{1}^{d}}⟨x − a⟩_{ℝ^{p}} = d.  (4.3)

Now, for each d ∈ D, let τ = τ(d) ≥ 0, ε = ε(d) > 0 and let J_{ετ}^{d} be the functional defined in V_{h} as J_{ετ} in (3.10) with L^{d} and Π^{d} instead of L and Π respectively. Finally, let σ_{ετ}^{dh} be the discrete smoothing variational spline with tangent conditions in Ω′ relative to A_{1}^{d}, A_{2}^{d}, f, τ and ε, which is the minimum of J_{ετ}^{d} in V_{h}.

To prove the convergence of σ_{ετ}^{dh} to f, under suitable hypotheses, we need the following results.

Proposition 4.1. Let B_{0} = {b_{01}, …, b_{0△}} be a P̃_{m−1}(Ω′; ℝ^{n})-unisolvent subset of points of R. Then, there exists η > 0 such that if B_{η} designs the set of △-uplet B = {b_{1},...,b_{△}} of points of Ω′ satisfying the condition: for j = 1, …, △ and ⟨b_{j} − b_{0j}⟩_{ℝ^{p}} < η, the application

[[v]]_{m,Ω′}^{B} = ( ∑_{j=1}^{△}⟨v(b_{j})⟩_{ℝ^{n}}^{2} + | v |_{m,Ω′,ℝ^{n}}^{2} )^{1/2},

defined, for all B ∈ B_{η}, is a norm on H^{m}(Ω′; ℝ^{n}), uniformly equivalent over B_{η} to the usual Sobolev norm ∥ · ∥_{m,Ω′,ℝ^{n}}.

The proof of this proposition is analogous to the proof of [1, proposition 6.2]. Now, we assume that the family (X_{h})_{h∈H} is such that there exists a linear

operator ρ_{h} : L^{2}(Ω′; ℝ^{3}) → V_{h} satisfying

(i) For all l = 0, …, m, and all y ∈ H^{m}(Ω′; ℝ^{3}),
|y − ρ_{h}y|_{l,Ω′,ℝ^{3}} ≤ Ch^{m−l}|y⃗|_{m,Ω′,ℝ^{3}};  (4.4)
(ii) For all y ∈ H^{m}(Ω′; ℝ^{3}),

lim_{h→0} |y − ρ_{h}y|_{m,Ω′,ℝ^{3}} = 0 .

Also assume that the cardinality of the subsets A_{1}^{d} and A_{2}^{d} satisfies

max{N_{1}(d), N_{2}(d)} ≤ C d^{−p},  (4.5)

and that the family (T_{h})_{h∈H} satisfies the inverse assumptions (c.f. P. G. Ciarlet [3]):

∃ν > 0, ∀h ∈ H, ∀K ∈ T_{h}, \frac{h}{h_{K}} ≤ ν.  (4.6)

Lemma 4.2. Assume that (4.2), (4.3), (4.4), (4.5), (4.6) hold. Then, there exists a constant C > 0 such that for any y ∈ H^{m}(Ω′; ℝ^{3}), d ∈ D and h ∈ H, one has

∑_{a∈A}⟨(ρ_{h}y − y)(a)⟩_{ℝ^{3}}^{2} ≤ Ch^{2m−2}|y|_{m,Ω′,ℝ^{3}}^{2},  (4.7)

∑_{b∈B^{d}}⟨D_{i}(ρ_{h}y − y)(b)⟩_{ℝ^{3}}^{2} ≤ C \frac{h^{2m−4}}{d^{2}} |y|_{m,Ω′,ℝ^{3}}^{2}, i = 1, 2 .  (4.8)

Proof. Reasoning as in [1, Lemma 6.1], we deduce that there exists a constant C > 0 such that for any y ∈ H^{m}(Ω; ℝ^{3}), d ∈ D and h ∈ H, and for any K ∈ T_{h}, one has

max_{u∈K}⟨y(u)⟩_{ℝ^{3}}^{2} ≤ Ch^{−2} ∑_{ℓ=0}^{m} h^{2ℓ}|y|_{ℓ,K,ℝ^{3}}^{2},  (4.9)

max_{u∈K}⟨D_{i}y(u)⟩_{ℝ^{3}}^{2} ≤ Ch^{−2} ∑_{ℓ=0}^{m−1} h^{2ℓ}|y|_{ℓ+1,K,ℝ^{3}}^{2}, i = 1, 2.  (4.10)

Thus, taking ρ_{h}y − y instead of y in (4.9), we deduce that

∑_{a∈A}⟨(ρ_{h}y − y)(a)⟩_{ℝ^{3}}^{2} ≤ ∑_{K∈T_{h}} ∑_{a∈A∩K}⟨(ρ_{h}y − y)(a)⟩_{ℝ^{3}}^{2}

≤ Ch^{−2}N_{1} ∑_{K∈T_{h}} ∑_{ℓ=0}^{m} h^{2ℓ}|ρ_{h}y − y|_{ℓ,K,ℝ^{3}}^{2}

≤ Ch^{−2}N_{1} ∑_{ℓ=0}^{m} h^{2ℓ}|ρ_{h}y − y|_{ℓ,Ω′,ℝ^{3}}^{2} .

Hence, from (4.4), we have

∑_{a∈A}⟨(ρ_{h}y − y)(a⟩_{ℝ^{3}}^{2} ≤ Ch^{−2}N_{1} ∑_{ℓ=0}^{m} h^{2ℓ}h^{2m−2ℓ}|y|_{m,Ω′,ℝ^{3}}^{2}
≤ Ch^{−2}N_{1}(m + 1)h^{2m}|y|_{m,Ω′,ℝ^{3}}^{2},

and we conclude that (4.7) holds. Analogously, taking ρ_{h}y −y instead of y in (4.10) we deduce, for i = 1, 2, that

∑_{b∈B^{d}}⟨D_{i}(ρ_{h}y − y)(b)⟩_{ℝ^{3}}^{2} ≤ Ch^{−2}N_{2} ∑_{ℓ=0}^{m−1} h^{2ℓ}|ρ_{h}y − y|_{ℓ+1,Ω′,ℝ^{3}}^{2}.

Hence, from (4.4), we have

∑_{b∈B^{d}}⟨D_{i}(ρ_{h}y − y⟩_{ℝ^{3}}^{2} ≤ Ch^{−2}N_{2}mh^{2m−2}|y|_{m,Ω′,ℝ^{3}}^{2},

and, from (4.5), we conclude that (4.8) holds. □

Now we state the main result.

Theorem 4.3. Assume that (4.2)–(4.6) hold, and that

ε = o(d^{−p}), d → 0,  (4.11)
\frac{τ h^{2m−4}}{d^{p} ε} = o(1), d → 0,  (4.12)
\frac{h^{2m}}{d^{p} ε} = o(1), d → 0,  (4.13)

Then, one has

lim_{d→0} ∥σ_{ετ}^{dh} − f∥_{m,Ω′,ℝ^{n}} = 0.  (4.14)

Proof. Step 1. For all d ∈ D, from Theorem 3.3 we have

J_{ετ}^{d} (σ_{ετ}^{dh}) ≤ J_{ετ}^{d} (ρ_{h}f)

where, for each h ∈ H, ρ_{h} is the operator given in (4.4), which means that

|σ_{ετ}^{dh}|_{m,Ω′,ℝ^{n}}^{2} ≤ \frac{1}{ε}⟨L^{d}(ρ_{h}f − f)⟩_{N_{1},n}^{2} + \frac{τ}{ε} ⟨Π^{d}(ρ_{h}f)⟩_{N_{2},pn}^{2} + |ρ_{h}f|_{m,Ω′,ℝ^{n}}^{2}.

From (4.4) and Lemma 4.2, we obtain

⟨L^{d}(ρ_{h}f − f)⟩_{N_{1},n}^{2} ≤ CN_{1}h^{2m}|f|_{m,Ω′,ℝ^{n}}^{2}

and taking into account

⟨Π^{d}(ρ_{h}f)⟩_{N_{2},pn}^{2} ≤ CpN_{2}h^{2m−4}|f|_{m,Ω′,ℝ^{n}}^{2},
|ρ_{h}f|_{m,Ω′,ℝ^{n}}^{2} = o(1) + |f|_{m,Ω′,ℝ^{n}}^{2}, d → 0 ,

we deduce from (4.5) and Lemma 4.2 that there exist C_{1} > 0 and C_{2} > 0 such that

|σ_{ετ}^{dh}|_{m,Ω′,ℝ^{n}}^{2} ≤ (\frac{C_{1}h^{2m}}{d^{p}ε} + \frac{C_{2}h^{2m−4}τ}{d^{p}ε} + 1)|f|_{m,Ω′,ℝ^{n}}^{2} + o(1), d → 0,  (4.15)
⟨L^{d}(σ_{ετ}^{dh} − f)⟩_{N_{1},n}^{2} = o(ε), as d → 0.  (4.16)

Let B_{0} = {b_{01}, …, b_{0Δ}} be a P̃_{m−1}(Ω′)-unisolvent subset of points of R and let η be the constant of the Proposition 4.1. Obviously, there exists η′ ∈ (0, η] such that

\overline{B}(b_{0j}, η′) ⊂ \overline{R} for j = 1, …, Δ .

From (4.3), for all d ∈ D, d < η′, j = 1, …, Δ,

\overline{B}(b_{0j}, η′ − d) ⊂ ⋃_{a∈A_{1}^{d}∩\overline{B}(b_{0j},η′)} \overline{B}(a, d) .

If N_{j} = card(A_{1}^{d} ∩ \overline{B}(b_{0j}, η′)), it follows: for all d ∈ D, d < η′, j = 1, …, Δ,

(η′ − d)^{p} ≤ N_{j}d^{p} .

Consequently, for any d_{0} ∈ (0, η′), all d ∈ D, d ≤ d_{0}, j = 1, …, Δ,

N_{j} ≥ (η′ − d_{0})^{p}d^{−p} .  (4.17)

Now, from (4.11) and (4.16) it follows that for j = 1, …, Δ,

∑_{a∈A_{1}^{d}∩\overline{B}(b_{0j},η′)}⟨(σ_{ετ}^{dh} − f)(a)⟩_{ℝ^{n}}^{2} = o(d^{−p}), as d → 0 .  (4.18)

If a_{j}^{d} is a point of A_{1}^{d} ∩ \overline{B}(b_{0j}, η′) such that

⟨(σ_{ετ}^{dh} − f)(a_{j}^{d})⟩_{ℝ^{n}} = min_{a∈A_{1}^{d}∩\overline{B}(b_{0j},η′)}⟨(σ_{ετ}^{dh} − f)(a)⟩_{ℝ^{n}} ,

it follows from (4.17) and (4.18) that for j = 1, …, Δ,

⟨(σ_{ετ}^{dh} − f)(a_{j}^{d})⟩_{ℝ^{n}} = o(1), as d → 0 .  (4.19)

We denote by B^{d} the set {a_{1}^{d}, …, a_{Δ}^{d}}. Applying the Proposition 4.1 with B = B^{d}, for d sufficiently close to 0, it results from (4.12), (4.13), (4.15) and (4.19) that There exists C > 0, α > 0, such that for all d ∈ D, d ≤ α we have

∥σ_{ετ}^{dh}∥_{m,Ω′,ℝ^{n}} ≤ C .

This implies that the family (σ_{ετ}^{dh})_{d∈D, d≤α} is bounded in H^{m}(Ω′; ℝ^{n}). Then, there exists a sequence (σ_{ε_{l}τ_{l}}^{d_{l}h_{l}})_{l∈ℕ}, extracted from such family, with lim_{l→+∞} d_{l} = 0,

lim_{l→+∞} h_{l} = 0, ε_{l} = ε(d_{l}), τ_{l} = τ(d_{l}), ε_{l} = o(d_{l}^{−p}), \frac{τ_{l} h_{l}^{2m−4}}{d_{l}^{p} ε_{l}} = o(1), \frac{h_{l}^{2m}}{d_{l}^{p} ε_{l}} = o(1), as l → +∞, and an element f^{∗} ∈ H^{m}(Ω′; ℝ^{n}) such that

f^{∗} converges weakly to σ_{ε_{l}τ_{l}}^{d_{l}h_{l}} inH^{m}(Ω′; ℝ^{n}) as l → +∞ .  (4.20)

Step 2. Arguing by contradiction, it is easy to prove that f^{∗} = f. Step 3. From (4.20) and taking into account that f^{∗} = f and H^{m}(Ω′; ℝ^{n}) is compactly injected in H^{m−1}(Ω′; ℝ^{n}) we have:

f = lim_{l→+∞} σ_{ε_{l}τ_{l}}^{d_{l}h_{l}} in H^{m−1}(Ω′; ℝ^{n}) .  (4.21)

Consequently,

lim_{l→+∞}((σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}, f))_{m−1,Ω′,ℝ^{n}} = ∥f∥_{m−1,Ω′,ℝ^{n}}^{2} .

Using again (4.20) and that f = f^{∗}, we obtain

lim_{l→+∞}(σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}, f)_{m,Ω′,ℝ^{n}} = lim_{l→+∞} (((σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}, f))_{m,Ω′,ℝ^{n}} − ((σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}, f))_{m−1,Ω′,ℝ^{n}})
= |f|_{m,Ω′,ℝ^{n}}^{2} .

(4.22)

Since, for all l ∈ ℕ,

|σ_{ε_{l}τ_{l}}^{d_{l}h_{l}} − f|_{m,Ω′,ℝ^{n}}^{2} = |σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}|_{m,Ω′,ℝ^{n}}^{2} + |f|_{m,Ω′,ℝ^{n}}^{2} − 2(σ_{ε_{l}τ_{l}}^{d_{l}h_{l}}, f)_{m,Ω′,ℝ^{n}} .

From (4.15) and (4.22) we deduce

lim_{l→+∞} |σ_{ε_{l}τ_{l}}^{d_{l}h_{l}} − f|_{m,Ω′,ℝ^{n}} = 0 ,

which, together with (4.21), imply

lim_{l→+∞} ∥σ_{ε_{l}τ_{l}}^{d_{l}h_{l}} − f∥_{m,Ω′,ℝ^{n}} = 0 .

Step 4. To complete this proof we will argue by contradiction. Suppose that (4.14) does’nt hold. Then, there exist a real number μ > 0 and three sequences (d_{l′})_{l′∈ℕ}, (h_{l′})_{l′∈ℕ}, (ε_{l′})_{l′∈ℕ}, and (τ_{l′})_{l′∈ℕ}, with

lim_{l′→+∞} d_{l′} = 0, h_{l′} = h(d_{l′}), ε_{l′} = ε(d_{l′}),

τ_{l′} = τ(d_{l′}), ε_{l′} = o(d_{l′}^{−p}),

\frac{τ_{l′}h_{l′}^{2m−4}}{d_{l′}^{p}ε_{l′}} = o(1), \frac{h_{l′}^{2m}}{d_{l′}^{p}ε}_{l′} = o(1),

as l′ → +∞, such that for all l′ ∈ ℕ,

∥σ_{ε_{l′} τ_{l′}}^{d_{l′} h_{l′}} − f∥_{m,Ω′,ℝ^{n}} ≥ μ .  (4.23)

Now, the sequence (σ_{ε_{l′} τ_{l′}}^{d_{l′} h_{l′}})_{l′∈ℕ} is bounded in H^{m}(Ω′; ℝ^{n}). Then, the reasoning of Steps 1–3 shows that there exists a subsequence convergent to f, which is a contradiction with (4.23). □

Acknowledgment. This work was partially supported by the Junta de Andalucía (Research group FQM/191).

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Abdelouahed Kouibia Centre d’Etudes Universitaire de Taza, Taza, Maroc

E-mail address: kouibia@ugr.es

Miguel Pasadas Departamento de Matemática Aplicada, ETSI Caminos, C. y P., Universidad de Granada, Severo Ochoa s/n, E-18071 Granada, Spain

E-mail address: mpasadas@ugr.es Fax (34) 958 243131