Effective construction of divergence-free wavelets on the square

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Effective construction of divergence-free wavelets on the

square

Souleymane Kadri Harouna, Valérie Perrier

To cite this version:

Souleymane Kadri Harouna, Valérie Perrier. Effective construction of divergence-free wavelets on

the square. Journal of Computational and Applied Mathematics, Elsevier, 2013, 240, pp.74-86.

�10.1016/j.cam.2012.07.029�. �hal-00727038�

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Effective construction of divergence-free wavelets

on the square

S. Kadri Harounaa_{, V. Perrier}b

a

INRIA Rennes - Bretagne Atlantique, Campus Universitaire de Beaulieu, 35042 Rennes cedex, France

b

Laboratoire Jean Kuntzmann, Universit´e de Grenoble et CNRS UMR 5224, B.P. 53, 38401 Grenoble cedex9, France

Abstract

We present an effective construction of divergence-free wavelets on the square, with suitable boundary conditions. Since 2D divergence-free vector functions are the curl of scalar stream-functions, we simply derive divergence-free multires-olution spaces and wavelets by considering the curl of standard biorthogonal multiresolution analyses (BMRAs) on the square. The key point of the theory is that the derivative of a 1D BMRA is also a BMRA, as established by Jouini and Lemari´e-Rieusset [11]. We propose such construction in the context of generic compactly supported wavelets, which allows fast algorithms. Examples illustrate the practicality of the method.

Keywords: Divergence-free wavelets, biorthogonal multiresolution analyses, Dirichlet boundary conditions

1. Introduction

Divergence-free wavelets on the whole space Rd_{have been firstly constructed}

by Battle-Federbush in the orthogonal case [3], and by Lemari´e-Rieusset in the biorthogonal one [15]. A first implementation for the analysis of incompressible vector fields was carried out by Urban [20]. More recently, Deriaz-Perrier pro-pose an alternative fast decomposition into divergence-free wavelets based on anisotropic (tensor-product) wavelets in the periodic case [7]. First extensions of these constructions to the hypercube Ω = [0, 1]d _{were published by}

Steven-son in [18, 19]. In particular [19] addresses the problem of finding bases of the divergence-free vector space, with free-slip boundary conditions:

Hdiv(Ω) = {u ∈ (L2(Ω))d : div(u) = 0 and u · ~ν = 0}. (1)

This space occurs in the resolution of incompressible Navier-Stokes equations in velocity-pressure formulation with Dirichlet boundary conditions for the velocity

Email addresses: souleymane.kadri_harouna@inria.fr(S. Kadri Harouna),

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at the boundary [9]. Having at hand a divergence-free basis and associated finite-dimensional approximation vector spaces of Hdiv(Ω) is the key-point for

developing new numerical methods for the direct simulation of turbulence, as proposed in [8] in the periodic case.

Our objective in the present paper is to propose, in the two-dimensional case, an alternative construction to [19] of wavelet bases of Hdiv(Ω), more generic and

effective. Indeed, unlike the construction [19], our bases follow from standard biorthogonal wavelets on the interval satisfying homogeneous boundary condi-tions, which include a wide variety of functions and will provide fast divergence-free wavelet decompositions. Our construction is based on two arguments:

Firstly in the 2D case, the space Hdiv(Ω) coincides with the curl of scalar

stream functions with vanishing boundary conditions [9]:

Hdiv(Ω) = {u = curl χ ; χ ∈ H01(Ω)}. (2)

Moreover in the simple case of a square domain, the curl is an isomorphism from H1

0(Ω) to Hdiv(Ω) [9], then wavelet bases can be simply constructed by

taking the curl of scalar wavelet bases of H1

0(Ω). The most common way to

construct such bases of H1

0(Ω) is to consider tensor products of one-dimensional

wavelets of H1

0(0, 1): these 1D wavelets arise from 1D multiresolution analyses

of H1_{(0, 1), (V}1

j), as constructed in a significant number of works [5, 6, 10], and

adapted to homogeneous boundary conditions by skipping one function at each boundary [4, 16, 17].

The second argument in our construction follows from the work of Jouini -Lemari´e-Rieusset [11], which states that the derivative spaces d

dxV 1

j = Vj0

consti-tute a multiresolution analysis of L2_{(0, 1), allowing fast wavelet transforms. We}

show in next section that in the context of compactly supported wavelets with polynomial reproduction, the spaces V0

j, with associated scaling functions and

wavelets can be simply constructed from the characteristics of the V1 j-spaces.

Furthermore our construction integrate homogeneous boundary conditions to the spaces V1

j.

At this point, our construction differs radically from [19], where, inversely, the construction of V1

j-spaces of H01(0, 1) of derives from the integration of 1D MRAs

of L2_{-functions with zero mean, and finally need to complete the divergence-free}

basis with directionally constant functions.

The outline of the paper is as follows: in Section 2 we construct new V0 j

spaces and wavelets by differentiating standard BMRAs of H1_{(0, 1), as predicted}

by Jouini-Lemari´e Rieusset theory. Section 3 introduces our finite dimensional divergence-free spaces and associated wavelets. Examples on wavelets and in-compressible vector flow analysis show the feasibility of the method.

2. Construction of BMRAs on [0,1] linked by differentiation 2.1. The fundamental construction of Jouini and Lemari´e-Rieusset

The existence of divergence-free wavelet bases on the whole space Rd_follows

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Proposition 1. Let (V1

j(R), ˜Vj1(R)) be a biorthogonal MRA of L2(R), with

compactly supported scaling functions (ϕ1,ϕ˜1) and wavelets (ψ1, ˜ψ1), such that ϕ1, ψ1∈ C1+ε_{for ε > 0. Then there exists a biorthogonal MRA (V}0

j (R), ˜Vj0(R)),

with associated scaling functions (ϕ0_,_ϕ_˜0_{) and wavelets (ψ}0_{, ˜}_ψ0_{), such that:}

(ϕ1₎′_{(x) = ϕ}0_{(x) − ϕ}0_{(x − 1) and (ψ}1₎′ _{= 4 ψ}0_.

The dual functions verify: Rx+1

x ϕ˜

1_{(t) dt = ˜}_ϕ0_{(x) and ( ˜}_ψ0₎′

= −4 ˜ψ1_.

From the generators (ϕ1_,_ϕ_˜1_{), (ϕ}0_,_ϕ_˜0_{) of Proposition 1, Jouini and}

Lemari´e-Rieusset prove the existence of two one-dimensional MRA of L2_{(0, 1), (V}1 j) and (V0 j) linked by differentiation [11]: d dxV 1 j = Vj0. (3)

Moreover the biorthogonal spaces should satisfy: ˜ Vj0 = H01(0, 1) ∩ Z x 0 ˜ Vj1= n f : f′ ∈ ˜Vj1 and f (0) = f (1) = 0 o . (4) Our objective in the rest of the section is to provide an effective construction of such multiresolution analyses, which enables vanishing boundary conditions, and fast wavelet algorithms for practical computations. We begin by recalling the basic setting of MRAs of H1_{(0, 1) and H}1

0(0, 1), then we propose a practical

computation of the spaces (V0 j, ˜Vj0).

2.2. Regular MRA (Vj1, ˜Vj1) of L2(0, 1) with polynomial reproduction (r, ˜r), and

associated MRA (VD

j , ˜VjD) of H01(0, 1)

The construction of spaces (V1

j, ˜Vj1) is classical and based on biorthogonal

multiresolution analyses on the interval reproducing polynomials up to some degree r − 1 in V1

j and ˜r− 1 in ˜Vj1 [5, 6, 10]: the principle is to start with

generators (ϕ1_,_ϕ_˜1_{), that are biorthogonal scaling functions of a BMRA on R.}

We suppose that ϕ1_{is C}1+ε_{, ε > 0, compactly supported on [n}

min, nmax] (nmin,

nmaxintegers), and reproduces polynomials up to degree r − 1:

0 ≤ ℓ ≤ r − 1, x ℓ ℓ! = +∞ X k=−∞ ˜ p1_ℓ(k) ϕ1(x − k), ∀ x ∈ R, (5) with ˜p1_ℓ(k) = hxℓ!ℓ,ϕ˜

1_{(x − k)i. Similarly, ˜}_ϕ1_{reproduces polynomials up to degree}

˜

r− 1 and we note p1

ℓ(k) = hx

ℓ

ℓ!, ϕ

1_{(x − k)i. For j sufficiently large, the spaces}

Vj1on [0, 1] have the structure:

Vj1= V 1,♭ j ⊕ V 1,int j ⊕ V 1,♯ j , (6)

where V_j1,int= span{ϕ1

j,k(x) = 2j/2ϕ1(2jx− k) ; k = k♭, 2j− k♯} is the space

generated by interior scaling functions whose supports are included into [δ♭

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δ♯

2j] ⊂ [0, 1] (δ♭, δ♯∈ N be two fixed parameters), and k♭= δ♭− nmin and k♯=

δ♯+ nmax. Moreover

V_j1,♭_{= span{Φ}1,♭_j,ℓ(x) = 2j/2Φ1,♭ℓ (2jx) ; ℓ = 0, · · · , r − 1},

V_j1,♯_{= span{Φ}1,♯_j,ℓ_{(1 − x) = 2}j/2Φ1,♯ℓ (2j(1 − x)) ; ℓ = 0, · · · , r − 1},

are the edge spaces, the edge scaling functions at the edge 0 being defined in order to preserve the polynomial reproduction (5) on the interval [0, 1]:

0 ≤ ℓ ≤ r − 1, Φ1,♭_ℓ (x) = k♭−1 X k=1−nmax ˜ p1ℓ(k) ϕ1(x − k) χ[0,+∞[. (7)

At the edge 1, the edge scaling functions Φ1,♯_j,ℓ are constructed on ] − ∞, 1] by symmetry, using the transform T f (x) = f (1 − x). In practice we have to choose j ≥ jmin where jmin is the smallest integer which verifies jmin >

log2[nmax− nmin+ δ♯+ δ♭] to ensure that the supports of edge scaling functions

at 0 do not intersect the supports of edge scaling functions at 1. The polynomial reproduction in V1

j is then satisfied since, for 0 ≤ ℓ ≤ r − 1

and x ∈ [0, 1] we have: 2j/2₍₂j_x)ℓ ℓ! = 2 j/2_Φ1,♭ ℓ (2 j_{x) +} 2j−k♯ X k=k♭ ˜ p1_ℓ(k) ϕ1j,k(x) + 2j/2Φ 1,♯ ℓ (2 j (1 − x)). (8) Similarly, the biorthogonal spaces ˜Vj1 are defined with the same structure,

al-lowing the polynomial reproduction up to degree ˜r− 1: ˜

V_j1_{= span{˜}Φ1,♭j,ℓ}ℓ=0,˜r−1⊕ ˜Vj1,int⊕ span{˜Φ 1,♯

j,ℓ}ℓ=0,˜r−1, (9)

where ˜V_j1,int= span{ ˜ϕ1_j,k ; k = ˜k♭, 2j− ˜k♯} is the space generated by interior

scaling functions ˜ϕ1_j,k(x) = 2j/2_ϕ_˜1₍₂j_x_{− k), whose supports are included into}

[δ˜♭

2j,1 −

˜ δ♯

2j] (˜δ♭, ˜δ♯∈ N be two parameters). The edge scaling functions at 0 are

defined by: 0 ≤ ℓ ≤ ˜r − 1, Φ˜1,♭_ℓ (x) = ˜ k♭−1 X k=1−˜nmax p1ℓ(k) ˜ϕ1(x − k) χ[0,+∞[. (10)

The equality between dimensions of V1

j and ˜Vj1 is obtained by adjusting the

parameters ˜k♭ = ˜δ♭− ñmin and ˜k♯ = ˜δ♯+ ñmax (with [ñmin,n˜max] = supp ˜ϕ1,

˜

nmin, ˜nmax integers) such that: ∆j = dim(Vj1) = dim( ˜Vj1) = 2j− (δ♭+ δ♯) −

(nmax− nmin) + 2r + 1. Remark that (δ♭, δ♯) remain ”free” parameters of the

construction (often chosen equal to 0 or 1). The last step of the construction lies in the biorthogonalization process of the basis functions, since edge scaling functions of V1

j and ˜Vj1 are no more biorthogonal [2, 6, 10, 17]. Finally, the

spaces (V1

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A MRA of H1

0(0, 1) can be simply defined from (Vj1) by VjD= Vj1∩H01(0, 1).

As described in [4, 16, 17], it suffices to remove the edge scaling functions Φ1,♭₀ defined in (7) at edge 0 and Φ1,♯₀ at edge 1 which leads to:

VjD = span{Φ1,♭j,ℓ ; ℓ = 1, r − 1} ⊕ V 1,int j ⊕ span{Φ 1,♯ j,ℓ ; ℓ = 1, r − 1}. To simplify, we denote by ϕD

j,k the scaling functions of VjD:

V_jD_{= span{ϕ}D_j,k ; k = 1, ∆j− 2}.

In such case, we also remove the edge functions ˜Φ1,♭0 and ˜Φ 1,♯

0 defined in (10)

from ˜V_j1, to adjust the dimension of the biorthogonal space: ˜ VjD= span{˜Φ 1,♭ j,ℓ}ℓ=1,˜r−1⊕ ˜Vj1,int⊕ span{˜Φ 1,♯ j,ℓ}ℓ=1,˜r−1 (11)

After a biorthogonalization process, we finally note: ˜

VjD= span{ ˜ϕDj,k ; k = 1, ∆j− 2}

and the spaces (VD

j , ˜VjD) form a biorthogonal MRA of H01(0, 1).

2.3. Construction of (V0

j , ˜Vj0) linked by differentiation /integration with (Vj1, ˜Vj1)

We now construct spaces (V0

j, ˜Vj0), that will be related to the spaces (Vj1, ˜Vj1)

of Section 2.2 by relations (3,4) of differentiation/integration. In practice, we follows the same structure of construction, starting from generators (ϕ0_,_ϕ_˜0₎

arising from Proposition 1. Recalling that _dxdϕ1(x) = ϕ0(x) − ϕ0(x − 1), ϕ0 has for compact support [nmin, nmax− 1], and reproduces polynomials up to degree

r− 2: 0 ≤ ℓ ≤ r − 2, x ℓ ℓ! = +∞ X k=−∞ ˜ p0_ℓ(k) ϕ0(x − k), (12) with ˜p0 ℓ(k) = hx ℓ ℓ!,ϕ˜0(x − k)i = ˜p1ℓ+1(k) − ˜p1ℓ+1(k − 1) for ℓ = 0, · · · , r − 2.

From Proposition 1, the scaling function ˜ϕ0_{(x) =}Rx+1

x ϕ˜

1_{(t)dt has for compact}

support [˜nmin− 1, ˜nmax], and reproduces polynomials up to degree ˜r. Noting

p0_ℓ(k) = hxℓ

ℓ!, ϕ

0_{(x − k)i, we have p}1

ℓ(k) = p0ℓ+1(k + 1) − p0ℓ+1(k) for ℓ = 1, · · · , ˜r.

Like V1

j, multiresolution spaces Vj0 are defined to ensure the polynomial

reproduction on [0, 1] up to degree r − 2: V0 j = V 0,♭ j ⊕ V 0,int j ⊕ V 0,♯ j

where Vj0,int= span{ϕ0j,k(x) = 2j/2ϕ0(2jx−k) ; k = k♭, 2j−k♯+1} is generated

by interior scaling functions whose supports are included into [δ♭

2j,1−

δ♯

2j] ⊂ [0, 1],

the parameters δ♭, δ♯being chosen equal to those arising from the construction

of V1

j . The edge spaces

V_j0,♭= span{Φ0,♭j,ℓ(x) = 2 j/2_Φ0,♭ ℓ (2 j x) ; ℓ = 0, · · · , r − 2}, V_j0,♯= span{Φ0,♯j,ℓ(1 − x) = 2 j/2_Φ0,♯ ℓ (2 j (1 − x)) ; ℓ = 0, · · · , r − 2},

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are generated by the left edge scaling functions: 0 ≤ ℓ ≤ r − 2, Φ0,♭_ℓ (x) = k♭−1 X k=2−nmax ˜ p0ℓ(k) ϕ0(x − k) χ[0,+∞[.

Biorthogonal spaces ˜V_j0are defined with similar structure, allowing a polynomial reproduction up to degree ˜r, but satisfying vanishing boundary conditions at 0 and 1, to preserve the commutation between the derivation and the multiscale projectors [11]. Then:

˜

Vj0= span{˜Φ0,♭j,ℓ}ℓ=1,˜r⊕ ˜Vj0,int⊕ span{˜Φ 1,♯

j,ℓ}ℓ=1,˜r, (13)

with ˜V_j0,int_{= span{ ˜}ϕ0

j,k; k = ˜k♭+1, 2j−˜k♯}, and ˜Φ0,♭ℓ =

P˜k♭

k=1−˜nmaxp

0

ℓ(k) ˜ϕ0kχ[0,+∞[,

for l = 1, . . . , ˜r, be the left edge scaling functions vanishing at 0, the right edge scaling functions ˜Φ0,♯_j,ℓ being constructed by symmetry. In next proposi-tion we will need to use the edge scaling funcproposi-tion non vanishing at 0, ˜Φ0,♭₀ = Pk˜♭

k=1−˜nmax ϕ˜

0

k χ[0,+∞[. In practice now j should satisfy j > jmin with:

jmin>max{log2[nmax− nmin+ δ♯+ δ♭+ 1], log2[˜nmax− ˜nmin+ ˜δ♯+ ˜δ♭+ 1]},

and a simple calculation shows that dim(V0

j) = dim( ˜Vj0) = ∆j− 1.

The following proposition proves that d dxV 1 j = Vj0and dxdV˜ 0 j ⊂ ˜Vj1. Proposition 2.

(i) The interior scaling functions of (V1

j, Vj0) and ( ˜Vj1, ˜Vj0) satisfy: d dx(ϕ 1 j,k) = 2j[ϕ0j,k−ϕ0j,k+1], d dx( ˜ϕ 0 j,k) = 2j[ ˜ϕ1j,k−1− ˜ϕ1j,k], for k♭≤ k ≤ 2j−k♯,

and dim(V_j0,int) = dim(V_j1,int) + 1, dim( ˜V_j0,int) = dim( ˜V_j1,int) − 1. (ii) The edge scaling functions of (V1

j, Vj0) satisfy, for ℓ = 1, . . . , r − 1: (Φ1,♭0 )′= −ϕ0k♭, (Φ 1,♭ ℓ ) ′ = Φ0,♭ℓ−1− ˜p1ℓ(k♭− 1) ϕ0k♭, (Φ1,♯₀ )′_{(1 − x) = ϕ}0 2−k♯(x), (Φ 1,♯ ℓ ) ′ (1 − x) = −Φ0,♯ℓ−1(1 − x) + ˜p 1 ℓ(2 − k♯) ϕ02−k♯(x),

whereas those of ( ˜V_j1, ˜V_j0) are linked by: for ℓ = 1, . . . , ˜r_{− 1,} ( ˜Φ0,♭_ℓ )′_{= ˜}_Φ1,♭ ℓ−1− p 0 ℓ(˜k♭) ˜ϕ1˜_k ♭ , ( ˜Φ 0,♯ ℓ (1 − x)) ′ = −˜Φ1,♯_ℓ−1(1 − x) + p0ℓ(2 − ˜k♯) ˜ϕ1_1−˜_k ♯.

Moreover, the function ˜Φ0,♭₀ verifies : ( ˜Φ0,♭₀ )′_{= − ˜}_ϕ1 ˜ k♭.

Proof. (i) follows directly from Proposition 1. (ii) comes from the definitions of edge scaling functions: we focus on the first line of equalities, at edge 0. We note in the following ϕi

k(x) = ϕi(x − k) (i = 0, 1).

For ℓ = 0, differentiating equation (7) in ]0, +∞[, one obtains (˜p10(k) = 1, ∀k):

(Φ1,♭0 ) ′ = k♭−1 X k=1−nmax (ϕ0k− ϕ0k+1) χ]0,+∞[ = ϕ01−nmax χ]0,+∞[− ϕ 0 k♭ = −ϕ 0 k♭,

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since supp ϕ0

1−nmax = [nmin− nmax+ 1, 0].

In the same way, for ℓ = 1, r − 1: (Φ1,♭ℓ ) ′ = k♭−1 X k=1−nmax ˜ p1_ℓ(k) (ϕ0k− ϕ0k+1) χ]0,+∞[ = k♭−1 X k=2−nmax [˜p1_ℓ_{(k) − ˜}p1_ℓ_{(k − 1)] ϕ}0_kχ]0,+∞[− p1ℓ(k♭− 1) ϕ0k♭.

From (12), since ˜p0_ℓ−1(k) = ˜p1_ℓ_{(k) − ˜}p1_ℓ_{(k − 1) we get:} (Φ1,♭_ℓ )′= k♭−1 X k=2−nmax ˜ p0_ℓ−1(k) ϕ0kχ]0,+∞[− ˜p1ℓ(k♭− 1) ϕ0k♭ = Φ 0,♭ ℓ−1− ˜p 1 ℓ(k♭− 1) ϕ0k♭.

The proof for edge scaling functions at edge 1 and in the biorthogonal spaces ˜

V1

j and ˜Vj0is obtained with similar arguments. ✷

For easy reading, the bases of (V1

j , ˜Vj1) and (Vj0, ˜Vj0), arising after the

biorthog-onalization will be now denoted by (ϕ1

j,k,ϕ˜1j,k)k=1,∆j and (ϕ

0

j,k,ϕ˜0j,k)k=1,∆j−1.

The following proposition proves that our constructed BMRAs take place in the theoretical framework [11].

Proposition 3. The biorthogonal spaces (V1

j, ˜Vj1) and (Vj0, ˜Vj0) verify relations

(3, 4): d dxV 1 j = Vj0 and ˜Vj0= H01∩ Rx 0 V˜ 1 j.

Proof. The inclusion d dxV

1

j ⊂ Vj0is straightforward according to Proposition 2.

The equality of dimensions between spaces leads to the first equality. Moreover Proposition 2 implies: Rx 0 V˜ 1 j = ˜Vj0⊕ span{2j/2Φ˜ 0,♭ 0 (2jx) − ˜Φ 0,♭ 0 (0)}. Since for j _{≥ j}min, ˜Vj0 ⊂ H01(0, 1) and 2j/2Φ 0,♭ 0 (2j) − Φ 0,♭ 0 (0) = −Φ 0,♭ 0 (0) 6= 0 we obtain

the second equality. ✷

This result induces the commutation between multiscale projectors and differen-tiation. Let P1

j be the oblique projector on Vj1parallel to ( ˜Vj1)⊥, ˜Pj1its adjoint,

and P0

j, ˜Pj0the biorthogonal projectors associated with (Vj0, ˜Vj0), we have [11]:

Proposition 4. (i) ∀ f ∈ H1_{(0, 1),} d dx◦ P 1 jf = Pj0◦ dxd f, (ii) ∀ f ∈ H1 0(0, 1), dxd ◦ ˜Pj0f = ˜Pj1◦ dxd f.

We now define the change of basis matrices between the spaces (d dxV 1 j ,dxd V˜ 0 j) and (V0

j, ˜Vj1), useful in numerical computations.

Definition 1. Let (L1

j, L0j, ˜L0j) be the change of basis matrices defined by:

d dxϕ 1 j,k= ∆j−1 X n=1 (L1j)k,n ϕ0j,n, d dxϕ˜ 0 j,k= ∆j X n=1 ( ˜L0_j)k,n ϕ˜1j,n, (14)

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and − Z x 0 ϕ0j,k= ∆j X m=1 (L0j)k,m ϕ1j,m. (15)

As we will see in Proposition 7, the matrices L0j and L1j will occur in the

com-putation of wavelet filters, and their explicit expression is given by the following proposition.

Proposition 5.

Let L0_{and L}1_{be the renormalized matrices: L}1

j= 2jL1and L0j = 2−jL0. Then,

L1 _{is a rectangular matrix of size ∆}

j× (∆j− 1) whose non zero elements are:

L1k,k−1= 1, L1k,k= −1, r + 1 ≤ k ≤ ∆j− r and for 2 ≤ k ≤ r: L1_1,r_{= −1, L}1_k,k−1= 1, L1k,r= −˜p1k−1(k♭− 1) L1_∆_j_,∆_j_−r= 1, L1∆j−k+1,∆j−k+1= −1, L 1 ∆j−k+1,∆j−r= ˜p 1 k−1(2j− k♯+ 1)

Similarly, L0_{is a rectangular matrix of size (∆}

j−1)×∆jwhose no zero elements

are: L0k,m= −1, L0k,∆j = −1, k + 1 ≤ m ≤ ∆j− r and r ≤ k ≤ ∆j− r and for 2 ≤ k ≤ r: L0_r,1= 1, L0k−1,k= −1, L0k−1,1 = ˜p1k−1(k♭− 1) L0_∆_j_−r,∆_j _{= −1, L}0_∆_j_−k+1,∆_j_−k+1= 1, L0∆j−k+1,∆j = −˜p 1 k−1(2j− k♯+ 1) 2.3.1. Wavelet spaces

We first recall the structure of wavelet spaces of the BMRA (V1

j, ˜Vj1), which is

classical, although different kinds of wavelets may be designed [2, 4, 5, 6, 17, 10]. As for the scaling functions construction, in V1

j+1 one must distinguish interior

wavelets ψ1

j,k and edge wavelets denoted by Ψ 1,♭

j,ℓ at edge 0 and Ψ 1,♯

j,ℓ at edge

1. The interior wavelets ψ1

j,k correspond to wavelets on R whose supports are

included into [ δ♭

2j+1,1 −

δ♯

2j+1]. Since the the support of ψ1 (wavelet on R) is

[nmin−˜nmax+1

2 ,

nmax−˜nmin+1

2 ], a simple calculation [6, 10] shows that it suffices

to take k = p♭, 2j− p♯− 1, where p♭and p♯are integers defined by:

p♭= ⌊ ˜ nmax+ k♭− 1 2 ⌋ and p♯= ⌊ k♯− ˜nmin+ 1 2 ⌋.

For the edge wavelets it is important to have good localization. An example, proposed by [10] and used in our numerical tests, is to construct edge-wavelets of smallest support, which gives :

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and Ψ1,♯_j,ℓ :=I− Pj1− Q 1,int j ϕ1_j+1,2j+1_−k_♯_−2ℓ, ℓ= 0, . . . , p♯− 1, (17)

with Q1,intj the biorthogonal projector onto interior wavelets:

Q1,intj (f ) :=

X

p♭≤k≤2j−p♯−1

hf, ˜ψ1_j,k_iψ_j,k1 Then, the biorthogonal wavelet spaces associated to V1

j are defined by Wj1 =

V1

j+1∩ ( ˜Vj1)⊥ and, for j ≥ jmin, have the form: Wj1= W 1,♭ j ⊕ W 1,int j ⊕ W 1,♯ j , with      W_j1,♭ = span{Ψ1,♭j,ℓ(x) = 2j/2Ψ 1,♭ ℓ (2jx) ; ℓ = 0, p♭− 1}, W_j1,int = span{ψ1 j,k= 2j/2ψ1(2jx− k) ; k = p♭, 2j− p♯− 1}, W_j1,♯ = span_{Ψ1,♯_j,ℓ_{(1 − x) = 2}j/2_Ψ1,♯ ℓ (2j(1 − x)) ; ℓ = 0, p♯− 1},

The biorthogonal spaces ˜W1

j are constructed in the same way, finally the wavelet

bases of the two spaces must to be biorthogonalized identically as the scal-ing functions. The resultscal-ing wavelet bases are denoted by {ψ1

j,k}k=1,2j and

{ ˜ψ_j,k1 }k=1,2j without distinction.

The objective now is to exhibit biorthogonal wavelets of W0

j and ˜Wj0, linked

to ψ1

j,k and ˜ψj,k1 by differentiation/integration. This is done by the following

proposition, established in the general framework by [11]: Proposition 6. Let (V1

j, ˜Vj1) and (Vj0, ˜Vj0) BMRAs satisfying proposition 3.

The wavelet spaces W0

j = Vj+10 ∩ ( ˜Vj0)⊥ and ˜Wj0 = ˜Vj+10 ∩ (Vj0)⊥ are linked to

the biorthogonal wavelet spaces associated to (V1

j, ˜Vj1) by: W_j0= d dxW 1 j and W˜j0= Z x 0 ˜ W_j1. (18) Moreover, let {ψ1

j,k}k=1,2j and { ˜ψ1_j,k}_k=1,2j be two biorthogonal wavelet bases of

Wj1 and ˜Wj1. Biorthogonal wavelet bases of Wj0 and ˜Wj0 are directly defined by:

ψ_j,k0 = 2−j(ψ1j,k) ′ and ψ˜0_j,k_{= −2}j Z x 0 ˜ ψ_j,k1 . (19) Interior wavelets ψ0

j,k(x) = 2j/2ψ0(2jx− k) in this definition correspond to

classical wavelets, ψ0 _{being a wavelet on R associated to the scaling function}

ϕ0 _{as in Proposition 1. On the other hand, in standard constructions [4, 5,}

6, 17, 10], the edge wavelets of W0

j, ˜Wj0 do not verify the relations (19). The

next proposition guarantees that this new edge wavelets preserve fast algorithms since they satisfy two-scale equations.

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Proposition 7. Let {ψ1

j,k}k=1,2j and { ˜ψ1_j,k}_k=1,2j be two biorthogonal wavelet

bases of Wj1 and ˜Wj1 associated respectively to filters G1j and ˜G1j:

ψ_j,k1 =X n (G1j)k,n ϕ1j+1,n and ψ˜1j,k= X n ( ˜G1_j)k,n ϕ˜1j+1,n.

Then there exist sparse matrices G0j and ˜G0j defined by:

G0_j = 2−jG1_jL1_j+1 and G˜0_j _{= −2}jG˜1_jL0T_j+1, (20) such that the wavelets ψ0

j,k and ˜ψ0j,k satisfy: ψ0j,k= X n (G0j)k,nϕ0j+1,n and ψ˜0j,k= X n ( ˜G0j)k,nϕ˜0j+1,n.

Remark 1. The above construction of wavelets ψ0

j,k and ˜ψ0j,k has two main

interests: their filters are directly accessible from those of ψ1

j,k and ˜ψ1j,k, and

there is no need for biorthogonalization as for classical constructions. A MAT-LAB code computing corresponding low and high pass filters for several class of wavelet families can be downloaded from [14].

Proof. From the definition of wavelets (19) and of the change of basis (defini-tion 1): 2j_ψ0 j,k= X n (G1 j)k,n(ϕ1j+1,n)′= X n,m (G1 j)k,n(L1j+1)n,mϕ0j+1,m = X m [G1 jL1j+1]k,mϕ0j+1,m, since ψ0 j,k= P

m(G0j)k,mϕ0j+1,m, we obtain G0j. ˜G0j is obtained similarly. ✷

Example 1. Figure 1 shows the plots of edge scaling functions and wavelets at 0 in V1

j (left), and Vj0 (right). The generators (ϕ1,ϕ˜1) used are biorthogonal

B-Splines with r = ˜r= 3, nmin = −1, nmax = 2, ˜nmin = −3 and ˜nmax = 4.

The ”free” integer parameters are chosen as δ♭= δ♯= 2 and ˜δ♭= ˜δ♯= 0.

3. Divergence-free scaling functions and wavelets on the square 3.1. Construction of divergence-free MRA of Hdiv(Ω)

Let Ω = [0, 1]2_{. The aim of the present section is to provide a}

divergence-free MRA and wavelet bases of the space Hdiv(Ω), introduced in (1). Since

Hdiv(Ω) is also equal to (2), our construction consists in taking the curl of a

regular MRA of the space H1

0(Ω). Such MRA is given by the tensor-product

VjD⊗ VjDof a regular MRA of H01(0, 1):VjD= Vj1∩ H01(0, 1), as constructed in

Section 2. With the notations VD

j = span{ϕDj,k ; k = 1, ∆j− 2}, and ψj,kD the

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Definition 2. For j ≥ jmin, the divergence-free scaling function spaces Vjdiv

are defined by:

Vjdiv= curl(VjD⊗ VjD) = span{Φdivj,k}, (21)

where the divergence-free scaling functions are given by1_:

Φdivj,k := 1 √ 2curl[ϕ D j,k1⊗ ϕ D j,k2] = 1 √ 2 ϕ D j,k1⊗ (ϕ D j,k2) ′ ,−(ϕDj,k1) ′ ⊗ ϕDj,k2 . (22)

The spaces Vdiv

j defined above constitute an increasing sequence in (L2(Ω))2:

Vdiv

j ⊂ Vdivj+1, of dimension: dim(Vdivj ) = dim(VjD)2= (∆j− 2)2. We will also

consider the more standard MRA Vj of (L2(Ω))2:

Vj = (Vj1⊗ Vj0) × (Vj0⊗ Vj1), (23)

V_j0 being the spaces defined in Section 2.3. By Proposition 3, these discrete spaces Vj preserve the divergence-free condition, as stated in [11]:

u ∈ (L2(Ω))2, div(u) = 0 ⇒ div[Pj(u)] = 0, (24)

where Pj = (Pj1⊗ Pj0,Pj0⊗ Pj1) is the biorthogonal projector on Vj. In the

same way, we introduce anisotropic divergence-free wavelets and wavelet spaces: Definition 3. For j1, j2 ≥ jmin, the anisotropic divergence-free wavelets and

wavelet spaces are defined by: Ψdiv,1_j,k := √ 1 4j2+ 1curl[ϕ D jmin,k1⊗ ψ D j2,k2] and W div,1 j = span{Ψ div,1 j,k }, Ψdiv,2_j,k := √ 1 4j1+ 1curl[ψ D j1,k1⊗ ϕ D jmin,k2] and W div,2 j = span{Ψ div,2 j,k }, Ψdiv,3_j,k := √ 1 4j1+ 4j2curl[ψ D j1,k1⊗ ψ D j2,k2] and W div,3 j = span{Ψ div,3 j,k }.

We now prove the main result of the article, that is (Vdiv

j )j≥jmin is a

multires-olution analysis of Hdiv(Ω).

Proposition 8. The divergence-free scaling function spaces Vdiv

j and wavelet

spaces Wdiv,εj for ε = 1, 2, 3, satisfy:

(i) Vdiv

jmin⊂ · · · ⊂ V

div

j ⊂ Vdivj+1⊂ · · · ⊂ Hdiv(Ω) and ∪Vdivj = Hdiv(Ω).

(ii) Vdiv j = Vdivjmin L jmin≤j1,j2≤j−1(⊕ε=1,2,3W div,ε j ).

(iii) The set {Φdiv jmin,k,Ψ

div,ε

j,k }j1,j2≥jmin, ε=1,2,3 is a Riesz basis of Hdiv(Ω).

1

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Then each vector function u of Hdiv(Ω) has a unique decomposition: u =X k cdivjmin,kΦ div jmin,k+ X j,k X ε=1,2,3 ddiv,ε_j,k Ψdiv,ε_j,k , (25)

with the norm-equivalence: kuk2 L2∼ P k|cdivjmin,k| 2₊P j,k P ε=1,2,3|d div,ε j,k |2.

Proof. (i) Let Vjbe the spaces defined in (23). Since the spaces Hdiv(Ω)∩Vj

provide a multiresolution analysis of Hdiv(Ω) [11], (i) is reduced to prove

Vdiv

j = Hdiv(Ω) ∩ Vj.

According to Proposition 2, we have Vdiv

j ⊂ Vj and Vdivj ⊂ Hdiv(Ω) by

con-struction. Conversely, let u ∈ Hdiv(Ω) ∩ Vj, and Pj the biorthogonal projector

on Vj. We are going to prove that u ∈ Vdivj . On one hand, as u ∈ Vj we have

u = Pj(u), on the other hand due to u ∈ Hdiv(Ω) we have u = curl (χ) with

χ∈ H1

0(Ω), and thus u = Pj[curl (χ)]. Since the spaces (VjD⊗VjD)j≥jmin form

a MRA of H1

0(Ω), we can decompose χ as:

χ= PjD(χ) + X j1,j2≥j QD1,J(χ) + QD2,J(χ) + QD3,J(χ) , (J = (j1, j2)) where PjD(χ) = X k ck ϕDj,k1⊗ ϕ D j,k2, Q D 2,J(χ) = X j1≥j X k d2j1,kψ D j1,k1⊗ ϕ D j,k2, QD1,J(χ) = X j2≥j X k d1j2,k ϕ D j,k1⊗ ψ D j2,k2, Q D 3,J(χ) = X j1,j2≥j X k d3j,kψDj1,k1⊗ ψ D j2,k2,

are the biorthogonal projectors on respectively VD

j ⊗ VjD, WjD1⊗ V D j , VjD⊗ WjD2 and WD j1 ⊗ W D

j2. Proposition 2 implies that:

curl [ϕDj,k1⊗ ψ D j2,k2] ∈ (V D j ⊗ Wj02) × (V 0 j ⊗ WjD2), hence: Pj(curl [ϕDj,k1 ⊗ ψ D

j2,k2]) = 0, and same for Pj(curl [ψ

D j1,k1 ⊗ ϕ

D j,k2]),

Pj(curl [ψ_jD₁_,k₁⊗ ψD_j₂_,k₂]). This leads to:

Pj(curl (χ)) = Pj(curl [PjD(χ)]) = curl [PjD(χ)].

By construction we have curl [PD

j (χ)] ∈ Vdivj , which implies u ∈ Vdivj and then

completes the proof: Vdiv

j = Hdiv(Ω) ∩ Vj.

(ii) The spaces VD

j form a MRA of H01(0, 1), and we can write:

V_jD_{⊗ V}_jD = (VjDmin j−1 M j1=jmin W_jD₁_{) ⊗ (V}_jD_min j−1 M j2=jmin W_jD₂). By definition of Vdiv j , we obtain: Vdiv_j _{= curl} 2 4(VjDmin⊗V D jmin) M jmin≤j1,j2≤j−1 [(VjDmin⊗W D j2) ⊕ (W D j1⊗V D jmin) ⊕ (W D j1 ⊗W D j2)] 3 5,

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which leads to Vdiv j = Vdivjmin LhL jmin≤j1,j2≤j−1 ⊕ε=1,2,3Wdiv,εj i . (iii) First the completeness of the family {Φdiv

jmin,k,Ψ

div,ε

j,k }j1,j2≥jmin, ε=1,2,3 is

ensured by points (i) and (ii). To prove the L2_{-stability of the basis, we use}

a vaguelette argument, which is common in the wavelet theory, and recalled for instance in [11]. By assumption on 1D-spaces, the divergence-free wavelets Ψdiv,ε_j,k are compactly supported, zero mean value and belong to the space Cǫ_,

for ε > 0: they constitute a vaguelette-familly, and the stability (leading to the norm-equivalence (25)) follows from the existence of a biorthogonal vavelet family for the Ψdiv,ε_j,k , given by Proposition 9 below. ✷ We now construct a set biorthogonal divergence-free scaling functions and wavelets. Let: ˜ Φdivjmin,k:= 1 √ 2 ˜ ϕD jmin,k1⊗ ˜γjmin,k2 −˜γjmin,k1⊗ ˜ϕ D jmin,k2 , ˜Ψdiv,1j,k := 1 √ 4j2+ 1 2j2_ϕ_˜D jmin,k1⊗ ¯ψ 0 j2,k2 −˜γjmin,k1⊗ ˜ψ D j2,k2 ˜ Ψdiv,2_j,k :=√ 1 4j1+ 1 ˜ ψD j1,k1⊗ ˜γjmin,k2 −2j1_ψ¯0 j1,k1⊗ ˜ϕ D jmin,k2 , ˜Ψdiv,3_j,k := √ 1 4j1+ 4j2 2j2_ψ˜D j1,k1⊗ ¯ψ 0 j2,k2 −2j1_ψ¯0 j1,k1⊗ ˜ψ D j2,k2 where ˜γjmin,k = − Rx 0 ϕ˜ D jmin,k and ¯ψ 0 j,k = −2−j Rx 0 ψ˜ D

j,k, these wavelets

corre-sponding to ˜ψ0

j,k of (19) excepted for the edge functions. The following

propo-sition is straightforward.

Proposition 9. The families nΦdiv

j,k,Ψdiv,1j,k ,Ψ div,2 j,k ,Ψ div,3 j,k ; j1, j2≥j, k o and n ˜Φdiv j,k, ˜Ψdiv,1j,k , ˜Ψ div,2 j,k , ˜Ψ div,3 j,k ; j1, j2 ≥j, k o are biorthogonal in (L2_(Ω))2_.

Example 2. Figure 2 shows the vector representation of divergence-free edge scaling functions curl [Φ1,♭₁ ⊗ Φ1,♭1 ], curl [Φ

1,♭ 2 ⊗ Φ

1,♭

2 ] and wavelets curl [Ψ 1,♭ 1 ⊗ Ψ1,♭1 ], curl [Ψ 1,♭ 2 ⊗ Ψ 1,♭

2 ], constructed from biorthogonal B-Spline generators

(ϕ1_,_ϕ_˜1_{) with r = ˜}_r_{= 3 used in example 1.}

3.2. Incompressible vector flow analysis

As example, we consider a velocity field u of resolution 5122_{, arising from a}

numerical simulation of lid driven cavity flow, computed in the approximation space Vdiv

9 (see [12]). We computed its divergence-free decomposition (25), on

divergence-free functions constructed with the B-Spline generators of Figure 1. Figure 3 shows the vector field u (left) and corresponding divergence-free co-efficients. As expected, significant wavelet coefficients (left) are localized near shear zones.

In a second experiment, we computed the nonlinear approximation error pro-vided by divergence-free wavelets, and compared it with this propro-vided by stan-dard wavelet decomposition of the velocity u, and also with this provided by standard wavelet decomposition of the vorticity ω = curlu (the vorticity is of-ten preferred in numerical schemes to the velocity, since in 2D it is a scalar quantity; however the difficulty here is to impose boundary conditions on the

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vorticity, whereas it is straightforward on the velocity [9]). Figure 4 (left) shows the relative L2_{-error between u (or ω) and its N -best terms approximation, in}

function of N (in %): in Vdiv

9 for u, in (V9D⊗ V90) × (V90⊗ V10) for u = (u1, u2)

and in (V1

9 ⊗ V91) for ω. Clearly the nonlinear error provided by the

divergence-free wavelets behaves as well (or better) as the classical wavelet approximation for u and ω. But its main advantage lies in the preservation of the divergence-free property through the approximation step, which was also the case in the ω-formulation (but with the disadvantage of boundary conditions for ω): this is illustrated in Figure 4 (right), which plots the divergence of the N -best term ap-proximation of u, with divergence-free wavelets, and with standard wavelets (the divergence has been computed with a Finite Difference scheme at grid points, which explains why it does not converge to exactly 0).

4. Conclusion

We have presented an effective construction of divergence-free MRAs and wavelets on the square. Our construction, based on MRA on the interval al-lowing polynomial reproduction, respects the theoretical framework established by Jouini and Lemari´e-Rieusset [11]. Moreover it incorporates homogeneous boundary conditions in the basis functions, which allows the representation of the divergence-free space, with a free-slip boundary condition, and constitutes an alternative to the basis constructed by Stevenson [18]. Associated fast wavelet transforms can been implemented easily, since the basis functions verify simple two-scale equations. First realizations have been successfully presented in this article, and in [13] with the Helmholtz decomposition of a vector flow. Since Hdiv(Ω) is already a curl-space in the cube [0, 1]3[1], our construction extends

readily to the 3D-case. Work on more complex problems are underway, such as the direct simulation of turbulence, this will be the subject of forthcoming papers.

Acknowledgements

V. Perrier acknowledges the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-BS01-014-01 (TOMMI).

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[3] G. Battle, P. Federbush, Divergence-free vector wavelets, Michigan Math. Journ. 40 (1993) 181–195.

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[8] E. Deriaz, V. Perrier, Direct Numerical Simulation of Turbulence using divergence-free wavelets, Multiscale Modeling & Simulation, SIAM 7(3) (2008) 1101–1129.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 φ1 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 φ1 1 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 φ1 2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0 0.5 1 1.5 ψ1 0 0 0.5 1 1.5 2 2.5 3 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 ψ1 1 0 0.5 1 1.5 2 2.5 3 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 ψ1 2 0 0.5 1 1.5 2 2.5 3 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 1: Left column: scaling functions Φ1_ℓ,♭(three first rows) and wavelets Ψ1_ℓ,♭(three last

rows). Right column: scaling functions Φ0_ℓ,♭ (three first rows) and wavelets Ψ0_ℓ,♭(three last

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0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Curl[φ1 1φ 1 1] at (0,0) 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Curl[φ1 2φ 1 2] at (0,0) 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Curl[ψ1 1ψ 1 1] at (0,0) 0 1 2 3 4 5 6 0 1 2 3 4 5 Curl[ψ1 2ψ 1 2] at (0,0)

Figure 2: Vector field of divergence-free scaling functions curl[Φ11,♭⊗Φ

1_,♭

1 ] and curl[Φ

1_,♭

2 ⊗

Φ12,♭], constructed from edge scaling functions: ℓ = 1, 2 (first line), Vector field of

divergence-free wavelets curl[Ψ11,♭⊗Ψ

1_,♭

1 ] and curl[Ψ

1_,♭

2 ⊗Ψ

1_,♭

2 ], constructed from edge wavelets: ℓ = 1, 2

(second line). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −5 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 60

Figure 3: Example of vector field 512x512 (left), its divergence-free scaling function coefficients (middle) and renormalized divergence-free wavelet coefficients (right).

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10−2 10−1 100 101 102 10−11 10−10 10−9 10−8 10−7 10−6 10−5 Ratio of Coefficients Relative L 2 norm Div−free Basis Standard Basis Vorticity 10−2 10−1 100 101 102 10−6 10−5 10−4 10−3 10−2 Ratio of Coefficients L2

Norm of the divergence

Div−free Basis Standard Basis

Figure 4: Left: relative nonlinear error versus the number of retained coefficients (in %):, computed on the velocity field of Figure 3, provided by divergence-free wavelets (green curve) and standard wavelets (blue), and the same for the vorticity in standard wavelets (red). Right: divergence of the N -best terms approximation of u in divergence-free wavelet approximation (green), and in standard vector wavelet approximation (blue).