# Three-Dimensional Biorthogonal Divergence-Free and Curl-Free Wavelets with Free-Slip Boundary

Authors: Yingchun Jiang; Qingqing Sun
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper deals with the construction of divergence-free and curl-free wavelets
on the unit cube, which satisfies the free-slip boundary conditions. First, interval wavelets
adapted to our construction are introduced. Then, we provide the biorthogonal divergence-free
and curl-free wavelets with free-slip boundary and simple structure, based on the characterization of corresponding spaces. Moreover, the bases are also stable.


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## Body

## 1. Introduction

In recent years, divergence-free and curl-free wavelets are generally studied, due to their potential use in many physical problems [1–5]. Anisotropic divergence-free and curl-free wavelets on the hypercube are firstly constructed in [6, 7], but all these functions only satisfy slip boundary conditions. However, the free-slip boundary is important in many cases, such as the solution of partial differential equations in incompressible fluids and electromagnetism. Inspired by this fact, [8, 9] give the construction of anisotropic divergence-free and curl-free wavelets with free-slip boundary, but the structure is very complicated and the basis functions are not explicit. Recently, based on a simple characterization of 2D divergence-free space, Harouna and Perrier proposed an alternative construction to [8] for divergence-free wavelets in two-dimensional case [10]. Following the similar but nontrivial line, we mainly study the anisotropic 3D divergence-free and curl-free wavelet bases with free-slip boundary in this paper. The traditional understanding that 3D curl-free wavelets are more difficult to construct than divergence-free wavelets is not always right, due to our procedure.In Section2, interval wavelets that we will use are introduced. Based on the spaces characterization, 3D biorthogonal divergence-free and curl-free wavelet bases are given in Sections 3 and 4, respectively.


## 2. Interval Wavelets on[0,1]

In this part, we will introduce the interval wavelets used in the subsequent construction.The existence of divergence-free and curl-free wavelets onRd follows from the following fundamental proposition [11].Proposition 1.
Let(Vj1(R),V~j1(R)) be a biorthogonal MRA of L2(R), with compactly supported scaling functions (φ1,φ~1) and wavelets (ψ1,ψ~1), such that φ1,ψ1∈C1+ɛ for ɛ>0. Then there exists a biorthogonal MRA (Vj0(R),V~j0(R)), with associated scaling functions (φ0,φ~0) and wavelets (ψ0,ψ~0), such that
(1)(φ1)′(x)=φ0(x)-φ0(x-1),(ψ1)′=4ψ0.
The dual functions verify ∫xx+1‍φ~1(t)dt=φ~0(x) and (ψ~0)′=-4ψ~1.Based on the above proposition, Jouini and Lemarié-Rieusset [12] proved the existence of two one dimensional MRAs of L2(0,1) linked by
(2)ddxVj1=Vj0,V~j0=H01(0,1)⋂‍∫0x‍V~j1={f:f′∈V~j1,f(0)=f(1)=0}.
In the following, we simply introduce the construction of these spaces. Suppose that φ1 in Proposition 1 is supported on [nmin,nmax](nmin,nmaxintegers) and reproduces polynomials up to degree r-1:
(3)0≤ℓ≤r-1,xℓℓ!=∑k=-∞+∞‍p~ℓ1(k)φ1(x-k),x∈R,
with p~ℓ1(k)=〈xℓ/ℓ!,φ~1(x-k)〉. Similarly, φ~1 is supported on [n~min,n~max] and reproduces polynomials up to degree r~-1.Forj being sufficiently large, the spaces Vj1 have the structure
(4)Vj1=span{Φj,ℓ1,♭=2j/2Φℓ1,♭(2jx)}ℓ=0,…,r-1⊕Vj1,int⊕span{Φj,ℓ1,♯=2j/2Φℓ1,♯(2jx)}ℓ=0,…,r-1,
where Vj1,int=span{φj,k1=2j/2φ1(2jx-k):k=k♭,…,2j-k♯} is the space whose supports are included into [δ♭/2j,1-δ♯/2j]⊂[0,1] (δ♭,δ♯∈N be two fixed parameters), and k♭=δ♭-nmin, k♯=δ♯+nmax. Moreover, Φℓ1,♭ are the edge scaling functions at the edge 0 being defined by
(5)Φℓ1,♭(x)=∑k=1-nmaxk♭-1‍p~ℓ1(k)φ1(x-k)χ[0,+∞).
At the edge 1, Φℓ1,♭ are defined by symmetry using Tf(x)=f(1-x).Similarly, the biorthogonal spacesV~j1 are defined with the same structure as
(6)V~j1=span{Φ~j,ℓ1,♭}ℓ=0,…,r~-1⊕V~j1,int⊕span{Φ~j,ℓ1,♯}ℓ=0,…,r~-1.
Adjusting the parameters such that
(7)Δj=dim(Vj1)=dim(V~j1)=2j-(δ♭+δ♯)-(nmax-nmin)+2r+1.
The last step of the construction is the biorthogonalization process, since the edge scaling functions of Vj1 and V~j1 are no more biorthogonal. Finally, (Vj1,V~j1) form a biorthogonal MRA of L2(0,1).As described in [13], removing the edge scaling functions Φ01,♭ and Φ01,♯ leads to
(8)VjD=span{Φj,ℓ1,♭}ℓ=1,…,r-1⊕Vj1,int⊕span{Φj,ℓ1,♯}ℓ=1,…,r-1=:span{φj,kD:k=1,…,Δj-2}.
Similarly, define V~jD=span{Φ~j,ℓ1,♭}ℓ=1,…,r~-1⊕V~j1,int⊕span{Φ~j,ℓ1,♯}ℓ=1,…,r~-1. After a biorthogonalization process, we finally note that
(9)V~jD=span{φ~j,kD:k=1,…,Δj-2},
and the spaces (VjD,V~jD) form a biorthogonal MRA of H01(0,1).The construction of(Vj0,V~j0) follows the same structure. Since (φ1)′(x)=φ0(x)-φ0(x-1), φ0 has compact support [nmin,nmax-1] and reproduces polynomials up to degree r-2:
(10)0≤ℓ≤r-2,xℓℓ!=∑k=-∞+∞‍p~ℓ0(k)φ0(x-k),
with p~ℓ0(k)=〈xℓ/ℓ!,φ~0(x-k)〉. The scaling function φ~0(x)=∫xx+1‍φ~1(t)dt has support [n~min-1,n~max] and reproduces polynomials up to degree r~. Consider
(11)Vj0=span{Φj,ℓ0,♭=2j/2Φℓ0,♭(2jx)}ℓ=0,…,r-2⊕Vj0,int⊕span{Φj,ℓ0,♯=2j/2Φℓ0,♯(2jx)}ℓ=0,…,r-2,
where Vj0,int=span{φj,k0=2j/2φ0(2jx-k):k=k♭,…,2j-k♯+1} and supports are included into [δ♭/2j,1-δ♯/2j]⊂[0,1]. The left edge scaling functions are
(12)Φℓ0,♭(x)=∑k=2-nmaxk♭-1‍p~ℓ0(k)φ0(x-k)χ[0,+∞).
Biorthogonal spaces V~j0 are similarly defined, but by satisfying vanishing boundary conditions at 0 and 1, then
(13)V~j0=span{Φ~j,ℓ0,♭}ℓ=1,…,r~⊕V~j0,int⊕span{Φ~j,ℓ0,♯}ℓ=1,…,r~,
with V~j0,int=span{φ~j,k0:k=k~♭+1,…,2j-k~♯} and Φ~ℓ0,♭=∑k=1-n~maxk~♭pℓ0(k)φ~0(x-k)χ[0,+∞) for ℓ=1,…,r~. It is easy to know dim(Vj0)=dim(V~j0)=Δj-1.In practice, we choosej≥jmin with
(14)jmin>max{log2[nmax-nmin+δ♯+δ♭+1],log2[n~max-n~min+δ~♯+δ~♭+1]}
to ensure that the supports of edge scaling functions at 0 do not intersect the supports of edge scaling functions at 1.The construction of wavelet spaces(Wj1,W~j1) can be seen from [13]. Moreover, they satisfy the following result.Proposition 2 (see [12]).
Let(Vj1,V~j1) and (Vj0,V~j0) be MRAs satisfying (d/dx)Vj1=Vj0 and V~j0=H01⋂‍∫0x‍V~j1; then the wavelet spaces Wj0 and W~j0 are linked to the biorthogonal wavelet spaces associated to (Vj1,V~j1) by
(15)Wj0=ddxWj1,W~j0=∫0x‍W~j1.
Moreover, let {ψj,k1}k=1,…,2j and {ψ~j,k1}k=1,…,2j be two biorthogonal wavelet bases of  Wj1 and W~j1. Biorthogonal wavelet bases of Wj0 and W~j0 are directly defined by
(16)ψj,k0=2-j(ψj,k1)′,ψ~j,k0=-2j∫0x‍ψ~j,k1.


## 3. Divergence-Free Wavelets on[0,1]3

LetΩ=[0,1]3 and let n→ be the normal vector; the boundary condition considered in [6] is u→·n→=0 on Γ=⋃k=13Γk with
(17)Γk=[0,1]k-1×{0}×[0,1]3-k,1≤k≤3.
It holds that u→·n→=0 on Γ if and only if uk=0 on Γk(1≤k≤3). We call it a slip boundary, which is shown in Figure 1.Figure 1
Slip  boundary  condition (divergence).In this section, we mainly consider the following space with free-slip boundary as Figure2(18)ℋdiv(Ω)={u→∈(L2(Ω))3:divu→=0,u→·n→=0on∂Ω}.Figure 2
Free-slip  boundary  condition (divergence).Foru→(x,y,z)=(u1,u2,u3)T, the 3D curl-operator is defined as
(19)curlu→=(∂2u3-∂3u2,∂3u1-∂1u3,∂1u2-∂2u1)T.Remark 3.
Taking Fourier transform on the both sides ofdivu→=0 leads to the equation
(20)ξ1u^1(ξ)+ξ2u^2(ξ)+ξ3u^3(ξ)=0,ξ=(ξ1,ξ2,ξ3).
In L2(R3), define the following functions
(21)φ^1(ξ)=ξ3u^2-ξ2u^3i(ξ12+ξ22+ξ32),φ^2(ξ)=ξ1u^3-ξ3u^1i(ξ12+ξ22+ξ32),φ^3(ξ)=ξ2u^1-ξ1u^2i(ξ12+ξ22+ξ32).
Then, according to (20), it is easy to verify that
(22)u^1(ξ)=i(ξ2φ^3(ξ)-ξ3φ^2(ξ)),u^2(ξ)=i(ξ3φ^1(ξ)-ξ1φ^3(ξ)),u^3(ξ)=i(ξ1φ^2(ξ)-ξ2φ^1(ξ)),
which is equivalent to u→=curlφ→. Therefore, any function u→∈(L2(R3))3 which satisfies divu→=0 can be characterized by curl operator as u→=curlφ→ with φ→∈(H1(R3))3. In fact, a similar result holds in 3D nonsmooth domains.Proposition 4 (see [14]).
There is a characterization(23)ℋdiv(Ω)={u→=curlφ→:φ→∈(H1(Ω))3,φ→×n→=0→on∂Ω}.Based on Proposition4, we give the following definition of divergence-free scaling function spaces.Definition 5.
Forj≥jmin, the divergence-free scaling function spaces V→jdiv are defined by
(24)V→jdiv=curl{(Vj0⊗VjD⊗VjD)×(VjD⊗Vj0⊗VjD)×(VjD⊗VjD⊗Vj0)}=span{Φj,kdiv,1,Φj,kdiv,2,Φj,kdiv,3},
where the divergence-free scaling functions are given by(25)Φj,kdiv,1=:12curl[(φj,k10·φj,k2D·φj,k3D,0,0)T]=12[φj,k10·φj,k2D·(φj,k3D)′δ2-φj,k10·(φj,k2D)′·φj,k3Dδ3],Φj,kdiv,2=:12curl[(0,φj,k1D·φj,k20·φj,k3D,0)T]=12[(φj,k1D)′·φj,k20·φj,k3Dδ3-φj,k1D·φj,k20·(φj,k3D)′δ1],Φj,kdiv,3=:12curl[(0,0,φj,k1D·φj,k2D·φj,k30)T]=12[φj,k1D·(φj,k2D)′·φj,k30δ1-(φj,k1D)′·φj,k2D·φj,k30δ2].
For proving the consequent main result, we also consider the standard MRAV→j of (L2(Ω))3:
(26)V→j=(Vj1⊗Vj0⊗Vj0)×(Vj0⊗Vj1⊗Vj0)×(Vj0⊗Vj0⊗Vj1).
The following conclusion shows that the spaceV→j preserves the divergence-free condition.Proposition 6.
Ifu→∈(L2(Ω))3 and divu→=0, then div[P→ju→]=0, where P→j=(pj1⊗pj0⊗pj0,pj0⊗pj1⊗pj0,pj0⊗pj0⊗pj1) is the biorthogonal projector on V→j.Proof.
Letu→=(u1,u2,u3)T; then
(27)P→ju→=(pj1⊗pj0⊗pj0u1,pj0⊗pj1⊗pj0u2,pj0⊗pj0⊗pj1u3)T.
Therefore, by the fact d/dx∘pj1f=pj0∘(d/dx)f in [10], we obtain
(28)div[P→ju→]=∂∂xpj1⊗pj0⊗pj0u1+∂∂ypj0⊗pj1⊗pj0u2+∂∂zpj0⊗pj0⊗pj1u3=pj0⊗pj0⊗pj0(∂u1∂x+∂u2∂y+∂u3∂z)=pj0⊗pj0⊗pj0(divu→)=0.Theorem 7.
The divergence-free scaling function spaces{V→jdiv}j≥jmin is a multiresolution analysis of ℋdiv(Ω).Proof.
Sinceℋdiv(Ω)∩V→j are a multiresolution analysis of ℋdiv(Ω), it is reduced to prove
(29)V→jdiv=ℋdiv(Ω)∩V→j.
Noting that (d/dx)Vj1=Vj0 and VjD⊆Vj1 from (2) and (8), we know V→jdiv⊂V→j. Furthermore, V→jdiv⊂ℋdiv(Ω) by construction. Therefore, V→jdiv⊂ℋdiv(Ω)∩V→j.
Conversely, lettingu→∈ℋdiv(Ω)∩V→j, we are going to prove u→∈V→jdiv. On the one hand, since u→∈V→j, we have u→=P→ju→. On the other hand, since u→∈ℋdiv(Ω), there exists a φ→=(φ1,φ2,φ3)T∈(H1(Ω))3 such that u→=curl(φ→). Moreover, φ→×n→=0→. Thus, u→=P→j[curl(φ→)]. Furthermore, we can decompose φ→ by isotropic vector wavelets as
(30)φ→=P→jD(φ→)+∑j′≥j‍Q→j′D(φ→),
where P→jD(φ→)=∑kc1,kφj,k10φj,k2Dφj,k3Dδ1+∑kc2,kφj,k1Dφj,k20φj,k3Dδ2+∑kc3,kφj,k1Dφj,k2Dφj,k30δ3(31)Q→j′D(φ→)=∑k‍(dj′,k1,1φj′,k10φj′,k2Dψj′,k3D+dj′,k1,2φj′,k10ψj′,k2Dφj′,k3D+dj′,k1,3ψj′,k10φj′,k2Dφj′,k3D+dj′,k1,4φj′,k10ψj′,k2Dψj′,k3D+dj′,k1,5ψj′,k10φj′,k2Dψj′,k3D+dj′,k1,6ψj′,k10ψj′,k2Dφj′,k3D+dj′,k1,7ψj′,k10ψj′,k2Dψj′,k3D)δ1+∑k‍(dj′,k2,1φj′,k1Dφj′,k20ψj′,k3D+dj′,k2,2φj′,k1Dψj′,k20φj′,k3D+dj′,k2,3ψj′,k1Dφj′,k20φj′,k3D+dj′,k2,4φj′,k1Dψj′,k20ψj′,k3D+dj′,k2,5ψj′,k1Dφj′,k20ψj′,k3D+dj′,k2,6ψj′,k1Dψj′,k20φj′,k3D+dj′,k2,7ψj′,k1Dψj′,k20ψj′,k3D)δ2+∑k‍(dj′,k3,1φj′,k1Dφj′,k2Dψj′,k30+dj′,k3,2φj′,k1Dψj′,k2Dφj′,k30+dj′,k3,3ψj′,k1Dφj′,k2Dφj′,k30+dj′,k3,4φj′,k1Dψj′,k2Dψj′,k30+dj′,k3,5ψj′,k1Dφj′,k2Dψj′,k30+dj′,k3,6ψj′,k1Dψj′,k2Dφj′,k30+dj′,k3,7ψj′,k1Dψj′,k2Dψj′,k30)δ3.
Since curl[φj′,k10φj′,k2Dψj′,k3Dδ1]∈(Vj′D⊗Vj′0⊗Wj′0)×(Vj′0⊗Vj′D⊗Wj′0)×(Vj′0⊗Vj′0⊗Wj′D), then
(32)P→j[curl(φj′,k10φj′,k2Dψj′,k3Dδ1)]=0→.
Similarly, every term in the right sides of (31) satisfies (32). Finally,
(33)P→j[curl(Q→j′D(φ→))]=0→.
Furthermore, we can obtain
(34)u→=P→j[curl(φ→)]=P→j[curl(P→jDφ→)]=curl(P→jDφ→).
Here, we have used the fact curl(P→jDφ→)∈V→j in the last step of (34). By construction, we have curl(P→jDφ→)∈V→jdiv, which means u→∈V→jdiv and the proof is completed.Based on the constructive method of vector wavelets and the following decompositions:(35)Vj0⊗VjD⊗VjD=(Vjmin0⊕j1=jminj-1Wj10)⊗(VjminD⊕j2=jminj-1Wj2D)⊗(VjminD⊕j3=jminj-1Wj3D)=(Vjmin0⊗VjminD⊗VjminD)⊕j3=jminj-1(Vjmin0⊗VjminD⊗Wj3D)⊕j2=jminj-1(Vjmin0⊗Wj2D⊗VjminD)⊕j1=jminj-1(Wj10⊗VjminD⊗VjminD)⊕j2,j3=jminj-1(Vjmin0⊗Wj2D⊗Wj3D)⊕j1,j3=jminj-1(Wj10⊗VjminD⊗Wj3D)⊕j1,j2=jminj-1(Wj10⊗Wj2D⊗VjminD)⊕j1,j2,j3=jminj-1(Wj10⊗Wj2D⊗Wj3D),VjD⊗Vj0⊗VjD=(VjminD⊗Vjmin0⊗VjminD)⊕j3=jminj-1(VjminD⊗Vjmin0⊗Wj3D)⊕j2=jminj-1(VjminD⊗Wj20⊗VjminD)⊕j1=jminj-1(Wj1D⊗Vjmin0⊗VjminD)⊕j2,j3=jminj-1(VjminD⊗Wj20⊗Wj3D)⊕j1,j3=jminj-1(Wj1D⊗Vjmin0⊗Wj3D)⊕j1,j2=jminj-1(Wj1D⊗Wj20⊗VjminD)⊕j1,j2,j3=jminj-1(Wj1D⊗Wj20⊗Wj3D),VjD⊗VjD⊗Vj0=(VjminD⊗VjminD⊗Vjmin0)⊕j3=jminj-1(VjminD⊗VjminD⊗Wj30)⊕j2=jminj-1(VjminD⊗Wj2D⊗Vjmin0)⊕j1=jminj-1(Wj1D⊗VjminD⊗Vjmin0)⊕j2,j3=jminj-1(VjminD⊗Wj2D⊗Wj30)⊕j1,j3=jminj-1(Wj1D⊗VjminD⊗Wj30)⊕j1,j2=jminj-1(Wj1D⊗Wj2D⊗Vjmin0)⊕j1,j2,j3=jminj-1(Wj1D⊗Wj2D⊗Wj30),
we can give the definition of anisotropic divergence-free wavelets as follows.Definition 8.
Forj1, j2, and j3≥jmin, the anisotropic divergence-free wavelets are defined by
(36)Ψj,kdiv,(1,1)=14j3+1curl[φjmin,k10·φjmin,k2D·ψj3,k3Dδ1]Ψj,kdiv,(1,2)=14j2+1curl[φjmin,k10·ψj2,k2D·φjmin,k3Dδ1]Ψj,kdiv,(1,3)=12curl[ψj1,k10·φjmin,k2D·φjmin,k3Dδ1]Ψj,kdiv,(1,4)=14j2+4j3curl[φjmin,k10·ψj2,k2D·ψj3,k3Dδ1]Ψj,kdiv,(1,5)=14j3+1curl[ψj1,k10·φjmin,k2D·ψj3,k3Dδ1]Ψj,kdiv,(1,6)=14j2+1curl[ψj1,k10·ψj2,k2D·φjmin,k3Dδ1]Ψj,kdiv,(1,7)=14j2+4j3curl[ψj1,k10·ψj2,k2D·ψj3,k3Dδ1],Ψj,kdiv,(2,1)=14j3+1curl[φjmin,k1D·φjmin,k20·ψj3,k3Dδ2]Ψj,kdiv,(2,2)=12curl[φjmin,k1D·ψj2,k20·φjmin,k3Dδ2]Ψj,kdiv,(2,3)=14j1+1curl[ψj1,k1D·φjmin,k20·φjmin,k3Dδ2]Ψj,kdiv,(2,4)=14j3+1curl[φjmin,k1D·ψj2,k20·ψj3,k3Dδ2]Ψj,kdiv,(2,5)=14j1+4j3curl[ψj1,k1D·φjmin,k20·ψj3,k3Dδ2]Ψj,kdiv,(2,6)=14j1+1curl[ψj1,k1D·ψj2,k20·φjmin,k3Dδ2]Ψj,kdiv,(2,7)=14j1+4j3curl[ψj1,k1D·ψj2,k20·ψj3,k3Dδ2],Ψj,kdiv,(3,1)=12curl[φjmin,k1D·φjmin,k2D·ψj3,k30δ3]Ψj,kdiv,(3,2)=14j2+1curl[φjmin,k1D·ψj2,k2D·φjmin,k30δ3]Ψj,kdiv,(3,3)=14j1+1curl[ψj1,k1D·φjmin,k2D·φjmin,k30δ3]Ψj,kdiv,(3,4)=14j2+1curl[φjmin,k1D·ψj2,k2D·ψj3,k30δ3]Ψj,kdiv,(3,5)=14j1+1curl[ψj1,k1D·φjmin,k2D·ψj3,k30δ3]Ψj,kdiv,(3,6)=14j1+4j2curl[ψj1,k1D·ψj2,k2D·φjmin,k30δ3]Ψj,kdiv,(3,7)=14j1+4j2curl[ψj1,k1D·ψj2,k2D·ψj3,k30δ3].Remark 9.
The coefficients before the operator “curl” are used to guarantee the biorthogonality in the following construction of dual wavelets.Proposition 10.
Defining the wavelet spaces(37)W→jdiv,(ɛ,n)=span{Ψj,kdiv,(ɛ,n)},ɛ=1,2,3,n=1,2,…,7;
then V→jdiv=V→jmindiv⊕jmin≤j1,j2,j3≤j-1(⊕ɛ=1,2,3,n=1,2,…,7W→jdiv,(ɛ,n)).Proof.
It can be easily obtained from (35) and Definition 8.Definition 11.
Biorthogonal divergence-free scaling functions and wavelets are defined by(38)Φ~j,kdiv,1=12[φ~j,k10·φ~j,k2D·γ~j,k3δ2-φ~j,k10·γ~j,k2·φ~j,k3Dδ3]Φ~j,kdiv,2=12[γ~j,k1·φ~j,k20·φ~j,k3Dδ3-φ~j,k1D·φ~j,k20·γ~j,k3δ1]Φ~j,kdiv,3=12[φ~j,k1D·γ~j,k2·φ~j,k30δ1-γ~j,k1·φ~j,k2D·φ~j,k30δ2],Ψ~j,kdiv,(1,1)=14j3+1[2j3φ~jmin,k10·φ~jmin,k2D·ψ~j3,k30δ2-φ~jmin,k10·γ~jmin,k2·ψ~j3,k3Dδ3]Ψ~j,kdiv,(1,2)=14j2+1[φ~jmin,k10·ψ~j2,k2D·γ~jmin,k3δ2-2j2φ~jmin,k10·ψ~j2,k20·φ~jmin,k3Dδ3]Ψ~j,kdiv,(1,3)=12[ψ~j1,k10·φ~jmin,k2D·γ~jmin,k3δ2-ψ~j1,k10·γ~jmin,k2·φ~jmin,k3Dδ3]Ψ~j,kdiv,(1,4)=14j2+4j3[2j3φ~jmin,k10·ψ~j2,k2D·ψ~j3,k30δ2-2j2φ~jmin,k10·ψ~j2,k20·ψ~j3,k3Dδ3]Ψ~j,kdiv,(1,5)=14j3+1[2j3ψ~j1,k10·φ~jmin,k2D·ψ~j3,k30δ2-ψ~j1,k10·γ~jmin,k2·ψ~j3,k3Dδ3]Ψ~j,kdiv,(1,6)=14j2+1[ψ~j1,k10·ψ~j2,k2D·γ~jmin,k3δ2-2j2ψ~j1,k10·ψ~j2,k20·φ~jmin,k3Dδ3]Ψ~j,kdiv,(1,7)=14j2+4j3[2j3ψ~j1,k10·ψ~j2,k2D·ψ~j3,k30δ2-2j2ψ~j1,k10·ψ~j2,k20·ψ~j3,k3Dδ3],Ψ~j,kdiv,(2,1)=14j3+1[γ~jmin,k1·φ~jmin,k20·ψ~j3,k3Dδ3-2j3φ~jmin,k1D·φ~jmin,k20·ψ~j3,k30δ1]Ψ~j,kdiv,(2,2)=12[γ~jmin,k1·ψ~j2,k20·φ~jmin,k3Dδ3-φ~jmin,k1D·ψ~j2,k20·γ~jmin,k3δ1]Ψ~j,kdiv,(2,3)=14j1+1[2j1ψ~j1,k10·φ~jmin,k20·φ~jmin,k3Dδ3-ψ~j1,k1D·φ~jmin,k20·γ~jmin,k3δ1]Ψ~j,kdiv,(2,4)=14j3+1[γ~jmin,k1·ψ~j2,k20·ψ~j3,k3Dδ3-2j3φ~jmin,k1D·ψ~j2,k20·ψ~j3,k30δ1]Ψ~j,kdiv,(2,5)=14j1+4j3[2j1ψ~j1,k10·φ~jmin,k20·ψ~j3,k3Dδ3-2j3ψ~j1,k1D·φ~jmin,k20·ψ~j3,k30δ1]Ψ~j,kdiv,(2,6)=14j1+1[2j1ψ~j1,k10·ψ~j2,k20·φ~jmin,k3Dδ3-ψ~j1,k1D·ψ~j2,k20·γ~jmin,k3δ1]Ψ~j,kdiv,(2,7)=14j1+4j3[2j1ψ~j1,k10·ψ~j2,k20·ψ~j3,k3Dδ3-2j3ψ~j1,k1D·ψ~j2,k20·ψ~j3,k30δ1],Ψ~j,kdiv,(3,1)=12[φ~jmin,k1D·γ~jmin,k2·ψ~j3,k30δ1-γ~jmin,k1·φ~jmin,k2D·ψ~j3,k30δ2]Ψ~j,kdiv,(3,2)=14j2+1[2j2φ~jmin,k1D·ψ~j2,k20·φ~jmin,k30δ1-γ~jmin,k1·ψ~j2,k2D·φ~jmin,k30δ2]Ψ~j,kdiv,(3,3)=14j1+1[ψ~j1,k1D·γ~jmin,k2·φ~jmin,k30δ1-2j1ψ~j1,k10·φ~jmin,k2D·φ~jmin,k30δ2]Ψ~j,kdiv,(3,4)=14j2+1[2j2φ~jmin,k1D·ψ~j2,k20·ψ~j3,k30δ1-γ~jmin,k1·ψ~j2,k2D·ψ~j3,k30δ2]Ψ~j,kdiv,(3,5)=14j1+1[ψ~j1,k1D·γ~jmin,k2·ψ~j3,k30δ1-2j1ψ~j1,k10·φ~jmin,k2D·ψ~j3,k30δ2]Ψ~j,kdiv,(3,6)=14j1+4j2[2j2ψ~j1,k1D·ψ~j2,k20·φ~jmin,k30δ1-2j1ψ~j1,k10·ψ~j2,k2D·φ~jmin,k30δ2]Ψ~j,kdiv,(3,7)=14j1+4j2[2j2ψ~j1,k1D·ψ~j2,k20·ψ~j3,k30δ1-2j1ψ~j1,k10·ψ~j2,k2D·ψ~j3,k30δ2].
Here, γ~j,k=-∫0x‍φ~j,kD(t)dt.Proposition 12.
The families{Φj,kdiv,ɛ,Ψj,kdiv,(ɛ,n):j1,j2,j3≥j,ɛ=1,2,3,n=1,2,…,7} and {Φ~j,kdiv,ɛ,Ψ~j,kdiv,(ɛ,n):j1,j2,j3≥j,ɛ=1,2,3,n=1,2,…,7} are biorthogonal in (L2(Ω))3.Proof.
It is easily proved by the fact thatψ~j,k0=-2j∫0x‍ψ~j,kD(t)dt, which is shown in (16).Theorem 13.
The set{Φjmin,kdiv,ɛ,Ψj,kdiv,(ɛ,n):j1,j2,j3≥jmin,ɛ=1,2,3,n=1,2,…,7} is a Riesz basis of ℋdiv(Ω).Proof.
The completeness is ensured by Theorem7 and Proposition 10. Now, it remains to prove the L2-stability of the basis. By assumption of 1D scaling and wavelet functions, the divergence-free wavelets Ψj,kdiv,(ɛ,n) are compactly supported, have zero mean value, and belong to the spaces Cɛ for some ɛ>0; then they constitute a vaguelette-family ([12]). Furthermore, the Riesz stability follows from the existence of a biorthogonal wavelet family given by Proposition 12.


## 4. Curl-Free Wavelets on[0,1]3

The boundary condition considered in [7] is u→×n→=0→ on Γ=⋃‍k=13Γk with
(39)Γk=⋃m=1,m≠k3‍[0,1]m-1×{0}×[0,1]3-m,1≤k≤3.It holds thatu→×n→=0→ on Γ if and only if uk=0 on Γk(1≤k≤3), which is shown in Figure 3.Figure 3
Slip  boundary  condition (curl).In this section, we mainly consider the following space:(40)ℋcurl(Ω)={u→∈(L2(Ω))3=:curlu→=0→,u→×n→=0→on∂Ω}
with free-slip boundary as Figure 4.Figure 4
Free-slip  boundary  condition (curl).An equivalent characterization is firstly given forℋcurl(Ω); and then we will give the MRA and wavelets for it.Proposition 14.
There is the characterizationℋcurl(Ω)={u→=gradφ:φ∈H01(Ω)}.Proof.
Supposeφ∈H01(Ω); then u→=gradφ=(∂1φ,∂2φ,∂3φ)T∈(L2(Ω))3. Moreover,
(41)curlu→=curl·gradφ=(∂2∂3φ-∂3∂2φ,∂3∂1φ-∂1∂3φ,∂1∂2φ-∂2∂1φ)T=0.→
Note that(42)∂1φ(x,y,0)=limΔx→0φ(x+Δx,y,0)-φ(x,y,0)Δx=0,∂1φ(x,y,1)=limΔx→0φ(x+Δx,y,1)-φ(x,y,1)Δy=0,∂1φ(x,0,z)=limΔx→0φ(x+Δx,0,z)-φ(x,0,z)Δx=0,∂1φ(x,1,z)=limΔx→0φ(x+Δx,1,z)-φ(x,1,z)Δy=0;
therefore,
(43)u1(x,y,0)=u1(x,y,1)=0,∀0≤x,y≤1.u1(x,0,z)=u1(x,1,z)=0,∀0≤x,z≤1.
In the same way, one can obtain
(44)u2(x,y,0)=u2(x,y,1)=0,∀0≤x,y≤1,u2(0,y,z)=u2(1,y,z)=0,∀0≤y,z≤1,u3(0,y,z)=u3(1,y,z)=0,∀0≤y,z≤1,u3(x,0,z)=u3(x,1,z)=0,∀0≤x,z≤1.
This is equivalent to u→×n→=0→. Therefore, u→=gradφ∈ℋcurl(Ω).
On the other hand, supposeu→∈ℋcurl(Ω); then we will prove that there exists a function φ∈H01(Ω), such that u→=gradφ. Since curlu→=0→, then
(45)∂2u3=∂3u2,∂3u1=∂1u3,∂1u2=∂2u1.
By Stokes formula, there exists a primitive function φ∈H1(Ω) such that
(46)dφ(x,y,z)=u1(x,y,z)dx+u2(x,y,z)dy+u3(x,y,z)dz.
Therefore, ∂φ/∂x=u1, ∂φ/∂y=u2, and ∂φ/∂z=u3; that is u→=gradφ. Furthermore, u→×n→=0→ means that
(47)u1(x,y,0)=u1(x,y,1)=0,∀0≤x,y≤1,u2(0,y,z)=u2(1,y,z)=0,∀0≤y,z≤1,u3(x,0,z)=u3(x,1,z)=0,∀0≤x,z≤1.
Noting that
(48)φ(x,y,z)=∫x0x‍u1(r,y,z)dr=∫y0y‍u2(x,s,z)ds=∫z0z‍u3(x,y,t)dt,
we obtain φ(x,y,0)=φ(x,y,1)=0, φ(0,y,z)=φ(1,y,z)=0, and φ(x,0,z)=φ(x,1,z)=0. Therefore, φ∈H01(Ω).Noting thatVjD⊗VjD⊗VjD is an MRA of (H01(Ω))3, we give the following definition.Definition 15.
Forj≥jmin, the curl-free scaling function spaces V→jcurl are defined by
(49)V→jcurl=grad(VjD⊗VjD⊗VjD)=span{Φj,kcurl},
where the curl-free scaling functions are given by
(50)Φj,kcurl=:13grad(φj,k1D·φj,k2D·φj,k3D)=13((φj,k1D)′·φj,k2D·φj,k3D,φj,k1D·(φj,k2D)′·φj,k3D,φj,k1D·φj,k2D·(φj,k3D)′)T.
For convenience, we also consider the standard MRAV→j of (L2(Ω))3:
(51)V→j=(Vj0⊗Vj1⊗Vj1)×(Vj1⊗Vj0⊗V11)×(Vj1⊗Vj1⊗Vj0).Theorem 16.
The curl-free scaling function spaces{V→jcurl}j≥jmin are a multiresolution analysis of ℋcurl(Ω).Proof.
Sinceℋcurl(Ω)∩V→j is a multiresolution analysis of ℋcurl(Ω), it is reduced to prove
(52)V→jcurl=ℋcurl(Ω)∩V→j.
Noting that (d/dx)Vj1=Vj0 and VjD⊆Vj1, we know V→jcurl⊂V→j. Furthermore, V→jcurl⊂ℋcurl(Ω) by construction. Therefore, V→jcurl⊂ℋcurl(Ω)∩V→j.
Conversely, letu→∈ℋcurl(Ω)∩V→j, we are going to prove u→∈V→jcurl. Let P→j be the biorthogonal projector on V→j. On the one hand, since u→∈V→j, we have u→=P→ju→. On the other hand, since u→∈ℋcurl(Ω), there exists a φ∈H01(Ω) such that u→=gradφ. Thus,
(53)u→=P→j[gradφ].
Since (VjD⊗VjD⊗VjD)j≥jmin forms an MRA of H01(Ω), we can decompose φ as
(54)φ=PjD(φ)+∑j1,j2,j3≥j‍∑n=17‍Qn,JD(φ),J=(j1,j2,j3),
where
(55)PjD(φ)=∑k‍ckφj,k1Dφj,k2Dφj,k3D,Q1,JD(φ)=∑j3≥j‍∑k‍dj3,k1φj,k1Dφj,k2Dψj3,k3D,Q2,JD(φ)=∑j2≥j‍∑k‍dj2,k2φj,k1Dψj2,k2Dφj,k3D,Q3,JD(φ)=∑j1≥j‍∑k‍dj1,k3ψj1,k1Dφj,k2Dφj,k3D,Q4,JD(φ)=∑j2,j3≥j‍∑k‍dj2,j3,k4φj,k1Dψj2,k2Dψj3,k3D,Q5,JD(φ)=∑j1,j3≥j‍∑k‍dj1,j3,k5ψj1,k1Dφj,k2Dψj3,k3D,Q6,JD(φ)=∑j1,j2≥j‍∑k‍dj1,j2,k6ψj1,k1Dψj2,k2Dφj,k3D,Q7,JD(φ)=∑j1,j2,j3≥j‍∑k‍dj1,j2,j3,k7ψj1,k1Dψj2,k2Dψj3,k3D
are the biorthogonal projectors on, respectively, VjD⊗VjD⊗VjD, VjD⊗VjD⊗Wj3D, VjD⊗Wj2D⊗VjD, Wj1D⊗VjD⊗VjD, VjD⊗Wj2D⊗Wj3D, Wj1D⊗VjD⊗Wj3D, Wj1D⊗Wj2D⊗VjD, and Wj1D⊗Wj2D⊗Wj3D.
Noting that(56)grad(φj,k1Dφj,k2Dψj3,k3D)∈(Vj0⊗VjD⊗Wj3D)×(VjD⊗Vj0⊗Wj3D)×(VjD⊗VjD⊗Wj30),
then P→j[gradQ1,JD(φ)]=0→. Similarly, P→j[gradQn,JD(φ)]=0→ for 2≤n≤7. Therefore,
(57)u→=P→j[gradφ]=P→j[gradPjD(φ)].
Since  gradPjD(φ)∈V→jcurl⊂V→j, then we obtain
(58)u→=gradPjD(φ)∈V→jcurl.Definition 17.
Forj1, j2, and j3≥jmin, the anisotropic curl-free wavelets and wavelet spaces are defined by
(59)Ψj,kcurl,1=14j3+2grad[φjmin,k1D·φjmin,k2D·ψj3,k3D]Ψj,kcurl,2=14j2+2grad[φjmin,k1D·ψj2,k2D·φjmin,k3D]Ψj,kcurl,3=14j1+2grad[ψj1,k1D·φjmin,k2D·φjmin,k3D]Ψj,kcurl,4=14j2+4j3+1grad[φjmin,k1D·ψj2,k2D·ψj3,k3D]Ψj,kcurl,5=14j1+4j3+1grad[ψj1,k1D·φjmin,k2D·ψj3,k3D]Ψj,kcurl,6=14j1+4j2+1grad[ψj1,k1D·ψj2,k2D·φjmin,k3D]Ψj,kcurl,7=14j1+4j2+4j3grad[ψj1,k1D·ψj2,k2D·ψj3,k3D].Proposition 18.
Defining the wavelet spacesW→jcurl,n=span{Ψj,kcurl,n} for n=1,2,…,7, then
(60)V→jcurl=V→jmincurl⊕jmin≤j1,j2,j3≤j-1(⊕ɛ=1,2,…,7W→jcurl,ɛ).Proof.
The result follows from the following fact:(61)VjD⊗VjD⊗VjD=(VjminD⊕j1=jminj-1Wj1D)⊗(VjminD⊕j2=jminj-1Wj2D)⊗(VjminD⊕j3=jminj-1Wj3D)=(VjminD⊗VjminD⊗VjminD)⊕j3=jminj-1(VjminD⊗VjminD⊗Wj3D)⊕j2=jminj-1(VjminD⊗Wj2D⊗VjminD)⊕j1=jminj-1(Wj1D⊗VjminD⊗VjminD)⊕j2,j3=jminj-1(VjminD⊗Wj2D⊗Wj3D)⊕j1,j3=jminj-1(Wj1D⊗VjminD⊗Wj3D)⊕j1,j2=jminj-1(Wj1D⊗Wj2D⊗VjminD)⊕j1,j2,j3=jminj-1(Wj1D⊗Wj2D⊗Wj3D).Definition 19.
Biorthogonal curl-free scaling functions and wavelets are defined by(62)Φ~j,kcurl=13[γ~j,k1·φ~j,k2D·φ~j,k3Dδ1+φ~j,k1D·γ~j,k2·φ~j,k3Dδ2+φ~j,k1D·φ~j,k2D·γ~j,k3δ3],Ψ~j,kcurl,1=14j3+2[γ~jmin,k1φ~jmin,k2Dψ~j3,k3Dδ1+φ~jmin,k1Dγ~jmin,k2ψ~j3,k3Dδ2+2j3φ~jmin,k1Dφ~jmin,k2Dψ~j3,k30δ3],Ψ~j,kcurl,2=14j2+2[γ~jmin,k1ψ~j2,k2Dφ~jmin,k3Dδ1+2j2φ~jmin,k1Dψ~j2,k20φ~jmin,k3Dδ2+φ~jmin,k1Dψ~j2,k2Dγ~jmin,k3δ3],Ψ~j,kcurl,3=14j1+2[2j1ψ~j1,k10φ~jmin,k2Dφ~jmin,k3Dδ1+ψ~j1,k1Dγ~jmin,k2φ~jmin,k3Dδ2+ψ~j1,k1Dφ~jmin,k2Dγ~jmin,k3δ3],Ψ~j,kcurl,4=14j2+4j3+1[γ~jmin,k1ψ~j2,k2Dψ~j3,k3Dδ1+2j2φ~jmin,k1Dψ~j2,k20ψ~j3,k3Dδ2+2j3φ~jmin,k1Dψ~j2,k2Dψ~j3,k30δ3],Ψ~j,kcurl,5=14j1+4j3+1[2j1ψ~j1,k10φ~jmin,k2Dψ~j3,k3Dδ1+ψ~j1,k1Dγ~jmin,k2ψ~j3,k3Dδ2+2j3ψ~j1,k1Dφ~jmin,k2Dψ~j3,k30δ3],Ψ~j,kcurl,6=14j1+4j2+1[2j1ψ~j1,k10ψ~j2,k2Dφ~jmin,k3Dδ1+2j2ψ~j1,k1Dψ~j2,k20φ~jmin,k3Dδ2+ψ~j1,k1Dψ~j2,k2Dγ~jmin,k3δ3],Ψ~j,kcurl,7=14j1+4j2+4j3[2j1ψ~j1,k10ψ~j2,k2Dψ~j3,k3Dδ1+2j2ψ~j1,k1Dψ~j2,k20ψ~j3,k3Dδ2+2j3ψ~j1,k1Dψ~j2,k2Dψ~j3,k30δ3].
Here, γ~j,k is defined as in Definition 11.Proposition 20.
The families{Φj,kcurl,Ψj,kcurl,ɛ:j1,j2,j3≥j,ɛ=1,2,…,7} and {Φ~j,kcurl,Ψ~j,kcurl,ɛ:j1,j2,j3≥j,ɛ=1,2,…,7} are biorthogonal in (L2(Ω))3.Theorem 21.
The set{Φjmin,kcurl,Ψj,kcurl,ɛ:j1,j2,j3≥jmin,ɛ=1,2,…,7} is a Riesz basis of ℋcurl(Ω).Proof.
It can be proved by the same method as Theorem13.



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*Source: 954717-2013-12-12.xml*