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

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.

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 Section 2, 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 on 𝑅^𝑑 follows from the following fundamental proposition [11].

Proposition 1. Let (𝑉¹_𝑗(𝑅),𝑉¹_𝑗(𝑅)) be a biorthogonal MRA of 𝐿²(𝑅), with compactly supported scaling functions (𝜑¹,𝜑¹) and wavelets (𝜓¹,𝜓¹), such that 𝜑¹,𝜓¹∈𝐶^1+𝜀 for 𝜀>0. Then there exists a biorthogonal MRA (𝑉⁰_𝑗(𝑅),𝑉⁰_𝑗(𝑅)), with associated scaling functions (𝜑⁰,𝜑⁰) and wavelets (𝜓⁰,𝜓⁰), such that 𝜑^1′(𝑥)=𝜑⁰(𝑥)−𝜑⁰(𝑥−1),𝜓^1′=4𝜓⁰. (1) The dual functions verify ∫_𝑥^𝑥+1𝜑¹(𝑡)𝑑𝑡=𝜑⁰(𝑥) and (𝜓⁰)^′=−4𝜓¹.

Based on the above proposition, Jouini and Lemarié-Rieusset [12] proved the existence of two one dimensional MRAs of 𝐿²(0,1) linked by 𝑑𝑉𝑑𝑥^1𝑗=𝑉⁰_𝑗,𝑉⁰_𝑗=𝐻¹₀(0,1)^𝑥₀𝑉¹_𝑗=𝑓∶𝑓^′∈𝑉¹_𝑗,𝑓(0)=𝑓(1)=0. (2) In the following, we simply introduce the construction of these spaces. Suppose that 𝜑¹ in Proposition 1 is supported on [𝑛_min,𝑛_max](𝑛_min,𝑛_maxintegers) and reproduces polynomials up to degree 𝑟−1: 𝑥0≤ℓ≤𝑟−1,^ℓ=ℓ!^+∞_𝑘=−∞𝑝¹_ℓ(𝑘)𝜑¹(𝑥−𝑘),𝑥∈𝑅, (3) with 𝑝¹_ℓ(𝑘)=⟨𝑥^ℓ/ℓ!,𝜑¹(𝑥−𝑘)⟩. Similarly, 𝜑¹ is supported on [𝑛_min,𝑛_max] and reproduces polynomials up to degree 𝑟̃−1.

For 𝑗 being sufficiently large, the spaces 𝑉¹_𝑗 have the structure 𝑉¹_𝑗=spanΦ^1,♭_𝑗,ℓ=2^𝑗/2Φ_ℓ^1,♭2^𝑗𝑥_{ℓ=0,…,𝑟−1}⊕𝑉_𝑗^1,int⊕spanΦ^1,♯_𝑗,ℓ=2^𝑗/2Φ_ℓ^1,♯2^𝑗𝑥_{ℓ=0,…,𝑟−1}, (4) where 𝑉_𝑗^1,int=span{𝜑¹_𝑗,𝑘=2^𝑗/2𝜑¹(2^𝑗𝑥−𝑘)∶𝑘=𝑘_♭,…,2^𝑗−𝑘_♯} is the space whose supports are included into [𝛿_♭/2^𝑗,1−𝛿_♯/2^𝑗]⊂[0,1] (𝛿_♭,𝛿_♯∈𝑁 be two fixed parameters), and 𝑘_♭=𝛿_♭−𝑛_min, 𝑘_♯=𝛿_♯+𝑛_max. Moreover, Φ_ℓ^1,♭ are the edge scaling functions at the edge 0 being defined by Φ_ℓ^1,♭(𝑥)=^𝑘♭−1_{𝑘=1−𝑛max}𝑝¹_ℓ(𝑘)𝜑¹(𝑥−𝑘)𝜒_[0,+∞). (5) At the edge 1, Φ_ℓ^1,♭ are defined by symmetry using 𝑇𝑓(𝑥)=𝑓(1−𝑥).

Similarly, the biorthogonal spaces 𝑉¹_𝑗 are defined with the same structure as 𝑉¹_𝑗=spanΦ^1,♭_{𝑗,ℓℓ=0,…,}𝑟̃₋₁⊕𝑉_𝑗^1,int⊕spanΦ^1,♯_{𝑗,ℓℓ=0,…,}𝑟̃₋₁. (6) Adjusting the parameters such that Δ_𝑗=dim𝑉¹_𝑗=dim𝑉¹_𝑗=2^𝑗−𝛿_♭+𝛿♯−𝑛_max−𝑛_min+2𝑟+1. (7) The last step of the construction is the biorthogonalization process, since the edge scaling functions of 𝑉¹_𝑗 and 𝑉¹_𝑗 are no more biorthogonal. Finally, (𝑉¹_𝑗,𝑉¹_𝑗) form a biorthogonal MRA of 𝐿²(0,1).

As described in [13], removing the edge scaling functions Φ₀^1,♭ and Φ₀^1,♯ leads to 𝑉^𝐷_𝑗=spanΦ^1,♭_{𝑗,ℓℓ=1,…,𝑟−1}⊕𝑉_𝑗^1,int⊕spanΦ^1,♯_{𝑗,ℓℓ=1,…,𝑟−1}=∶span𝜑^𝐷_𝑗,𝑘∶𝑘=1,…,Δ_𝑗−2. (8) Similarly, define 𝑉^𝐷_𝑗=span{Φ^1,♭_𝑗,ℓ}_ℓ=1,…,𝑟̃₋₁⊕𝑉_𝑗^1,int⊕span{Φ^1,♯_𝑗,ℓ}_ℓ=1,…,𝑟̃₋₁. After a biorthogonalization process, we finally note that 𝑉^𝐷_𝑗=span𝜑^𝐷_𝑗,𝑘∶𝑘=1,…,Δ_𝑗−2, (9) and the spaces (𝑉^𝐷_𝑗,𝑉^𝐷_𝑗) form a biorthogonal MRA of 𝐻¹₀(0,1).

The construction of (𝑉⁰_𝑗,𝑉⁰_𝑗) follows the same structure. Since (𝜑¹)^′(𝑥)=𝜑⁰(𝑥)−𝜑⁰(𝑥−1), 𝜑⁰ has compact support [𝑛_min,𝑛_max−1] and reproduces polynomials up to degree 𝑟−2: 𝑥0≤ℓ≤𝑟−2,^ℓ=ℓ!^+∞_𝑘=−∞𝑝⁰_ℓ(𝑘)𝜑⁰(𝑥−𝑘), (10) with 𝑝⁰_ℓ(𝑘)=⟨𝑥^ℓ/ℓ!,𝜑⁰(𝑥−𝑘)⟩. The scaling function 𝜑⁰(𝑥)=∫_𝑥^𝑥+1𝜑¹(𝑡)𝑑𝑡 has support [𝑛_min−1,𝑛_max] and reproduces polynomials up to degree 𝑟̃. Consider 𝑉⁰_𝑗=spanΦ^0,♭_𝑗,ℓ=2^𝑗/2Φ_ℓ^0,♭2^𝑗𝑥_{ℓ=0,…,𝑟−2}⊕𝑉_𝑗^0,int⊕spanΦ^0,♯_𝑗,ℓ=2^𝑗/2Φ_ℓ^0,♯2^𝑗𝑥_{ℓ=0,…,𝑟−2}, (11) where 𝑉_𝑗^0,int=span{𝜑⁰_𝑗,𝑘=2^𝑗/2𝜑⁰(2^𝑗𝑥−𝑘)∶𝑘=𝑘_♭,…,2^𝑗−𝑘_♯+1} and supports are included into [𝛿_♭/2^𝑗,1−𝛿_♯/2^𝑗]⊂[0,1]. The left edge scaling functions are Φ_ℓ^0,♭(𝑥)=^𝑘♭−1_{𝑘=2−𝑛max}𝑝⁰_ℓ(𝑘)𝜑⁰(𝑥−𝑘)𝜒_[0,+∞). (12) Biorthogonal spaces 𝑉⁰_𝑗 are similarly defined, but by satisfying vanishing boundary conditions at 0 and 1, then 𝑉⁰_𝑗=spanΦ^0,♭_{𝑗,ℓℓ=1,…,}𝑟̃⊕𝑉_𝑗^0,int⊕spanΦ^0,♯_{𝑗,ℓℓ=1,…,}𝑟̃, (13) with 𝑉_𝑗^0,int=span{𝜑⁰_𝑗,𝑘∶𝑘=𝑘_♭+1,…,2^𝑗−𝑘_♯} and Φ_ℓ^0,♭𝑘=∑^♭_𝑘=1−𝑛_max𝑝⁰_ℓ(𝑘)𝜑⁰(𝑥−𝑘)𝜒_[0,+∞) for ℓ=1,…,𝑟̃. It is easy to know dim(𝑉⁰_𝑗)=dim(𝑉⁰_𝑗)=Δ_𝑗−1.

In practice, we choose 𝑗≥𝑗_min with 𝑗_min>maxlog₂𝑛_max−𝑛_min+𝛿_♯+𝛿_♭+1,log₂𝑛_max−𝑛_min+𝛿_♯+𝛿_♭+1 (14) 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 (𝑊¹_𝑗,𝑊¹_𝑗) can be seen from [13]. Moreover, they satisfy the following result.

Proposition 2 (see [12]). Let (𝑉¹_𝑗,𝑉¹_𝑗) and (𝑉⁰_𝑗,𝑉⁰_𝑗) be MRAs satisfying (𝑑/𝑑𝑥)𝑉¹_𝑗=𝑉⁰_𝑗 and 𝑉⁰_𝑗=𝐻¹₀⋂∫^𝑥₀𝑉¹_𝑗; then the wavelet spaces 𝑊⁰_𝑗 and 𝑊⁰_𝑗 are linked to the biorthogonal wavelet spaces associated to (𝑉¹_𝑗,𝑉¹_𝑗) by 𝑊⁰_𝑗=𝑑𝑊𝑑𝑥^1𝑗,𝑊⁰_𝑗=^𝑥₀𝑊¹_𝑗. (15) Moreover, let {𝜓¹_𝑗,𝑘}_{𝑘=1,…,2𝑗} and {𝜓¹_𝑗,𝑘}_{𝑘=1,…,2𝑗} be two biorthogonal wavelet bases of  𝑊¹_𝑗 and 𝑊¹_𝑗. Biorthogonal wavelet bases of 𝑊⁰_𝑗 and 𝑊⁰_𝑗 are directly defined by 𝜓⁰_𝑗,𝑘=2^−𝑗𝜓¹_𝑗,𝑘^′,𝜓⁰_𝑗,𝑘=−2^𝑗𝑥₀𝜓¹_𝑗,𝑘. (16)

3. Divergence-Free Wavelets on [0,1]³

Let Ω=[0,1]³ and let 𝑛 be the normal vector; the boundary condition considered in [6] is 𝑢⋅𝑛=0 on Γ=⋃³_𝑘=1Γ_𝑘 with Γ_𝑘=[0,1]^𝑘−1×{0}×[0,1]^3−𝑘,1≤𝑘≤3. (17) It holds that 𝑢⋅𝑛=0 on Γ if and only if 𝑢_𝑘=0 on Γ_𝑘(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 Figure 2ℋ_div(Ω)=𝑢∈𝐿²(Ω)³∶div𝑢=0,𝑢⋅𝑛=0𝑜𝑛𝜕Ω. (18)

Figure 2 

Free-slip boundary condition (divergence).

For 𝑢(𝑥,𝑦,𝑧)=(𝑢₁,𝑢₂,𝑢₃)^𝑇, the 3D curl-operator is defined as curl𝑢=𝜕₂𝑢₃−𝜕₃𝑢₂,𝜕₃𝑢₁−𝜕₁𝑢₃,𝜕₁𝑢₂−𝜕₂𝑢₁^𝑇. (19)

Remark 3. Taking Fourier transform on the both sides of div𝑢=0 leads to the equation 𝜉₁𝑢₁(𝜉)+𝜉₂𝑢₂(𝜉)+𝜉₃𝑢₃(𝜉)=0,𝜉=𝜉₁,𝜉₂,𝜉₃. (20) In 𝐿²(𝑅³), define the following functions 𝜑₁(𝜉)=𝜉₃𝑢₂−𝜉₂𝑢₃𝑖𝜉²₁+𝜉²₂+𝜉²₃,𝜑₂(𝜉)=𝜉₁𝑢₃−𝜉₃𝑢₁𝑖𝜉²₁+𝜉²₂+𝜉²₃,𝜑₃(𝜉)=𝜉₂𝑢₁−𝜉₁𝑢₂𝑖𝜉²₁+𝜉²₂+𝜉²₃. (21) Then, according to (20), it is easy to verify that 𝑢₁(𝜉)=𝑖𝜉₂𝜑₃(𝜉)−𝜉₃𝜑₂(𝜉),𝑢₂(𝜉)=𝑖𝜉₃𝜑₁(𝜉)−𝜉₁𝜑₃(𝜉),𝑢₃(𝜉)=𝑖𝜉₁𝜑₂(𝜉)−𝜉₂𝜑₁(𝜉), (22) 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 ℋ_div(Ω)=𝑢=curl𝜑∶𝜑∈𝐻¹(Ω)³,𝜑×𝑛=0𝑜𝑛𝜕Ω. (23)

Based on Proposition 4, we give the following definition of divergence-free scaling function spaces.

Definition 5. For 𝑗≥𝑗_min, the divergence-free scaling function spaces 𝑉_𝑗^div are defined by 𝑉_𝑗^div=curl𝑉⁰_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗×𝑉^𝐷_𝑗⊗𝑉⁰_𝑗⊗𝑉^𝐷_𝑗×𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉⁰_𝑗=spanΦ^div,1_𝑗,𝐤,Φ^div,2_𝑗,𝐤,Φ^div,3_𝑗,𝐤, (24) where the divergence-free scaling functions are given byΦ^div,1_𝑗,𝐤1=∶√2curl𝜑⁰_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3,0,0^𝑇=1√2𝜑⁰_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3^′𝛿₂−𝜑⁰_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2^′⋅𝜑^𝐷_𝑗,𝑘3𝛿₃,Φ^div,2_𝑗,𝐤1=∶√2curl0,𝜑^𝐷_𝑗,𝑘1⋅𝜑⁰_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3,0^𝑇=1√2𝜑^𝐷_𝑗,𝑘1^′⋅𝜑⁰_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3𝛿₃−𝜑^𝐷_𝑗,𝑘1⋅𝜑⁰_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3^′𝛿₁,Φ^div,3_𝑗,𝐤1=∶√2curl0,0,𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑⁰_𝑗,𝑘3^𝑇=1√2𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2^′⋅𝜑⁰_𝑗,𝑘3𝛿₁−𝜑^𝐷_𝑗,𝑘1^′⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑⁰_𝑗,𝑘3𝛿₂. (25)
For proving the consequent main result, we also consider the standard MRA 𝑉_𝑗 of (𝐿²(Ω))³: 𝑉_𝑗=𝑉¹_𝑗⊗𝑉⁰_𝑗⊗𝑉⁰_𝑗×𝑉⁰_𝑗⊗𝑉¹_𝑗⊗𝑉⁰_𝑗×𝑉⁰_𝑗⊗𝑉⁰_𝑗⊗𝑉¹_𝑗. (26)
The following conclusion shows that the space 𝑉_𝑗 preserves the divergence-free condition.

Proposition 6. If 𝑢∈(𝐿²(Ω))³ and div𝑢=0, then div[𝑃_𝑗𝑢]=0, where 𝑃_𝑗=(𝑝¹_𝑗⊗𝑝⁰_𝑗⊗𝑝⁰_𝑗,𝑝⁰_𝑗⊗𝑝¹_𝑗⊗𝑝⁰_𝑗,𝑝⁰_𝑗⊗𝑝⁰_𝑗⊗𝑝¹_𝑗) is the biorthogonal projector on 𝑉_𝑗.

Proof. Let 𝑢=(𝑢₁,𝑢₂,𝑢₃)^𝑇; then 𝑃_𝑗𝑢=𝑝¹_𝑗⊗𝑝⁰_𝑗⊗𝑝⁰_𝑗𝑢₁,𝑝⁰_𝑗⊗𝑝¹_𝑗⊗𝑝⁰_𝑗𝑢₂,𝑝⁰_𝑗⊗𝑝⁰_𝑗⊗𝑝¹_𝑗𝑢₃^𝑇. (27) Therefore, by the fact 𝑑/𝑑𝑥∘𝑝¹_𝑗𝑓=𝑝⁰_𝑗∘(𝑑/𝑑𝑥)𝑓 in [10], we obtain div𝑃_𝑗𝑢=𝜕𝑝𝜕𝑥^1𝑗⊗𝑝⁰_𝑗⊗𝑝⁰_𝑗𝑢₁+𝜕𝑝𝜕𝑦^0𝑗⊗𝑝¹_𝑗⊗𝑝⁰_𝑗𝑢₂+𝜕𝑝𝜕𝑧^0𝑗⊗𝑝⁰_𝑗⊗𝑝¹_𝑗𝑢₃=𝑝⁰_𝑗⊗𝑝⁰_𝑗⊗𝑝⁰_𝑗𝜕𝑢₁+𝜕𝑥𝜕𝑢₂+𝜕𝑦𝜕𝑢₃𝜕𝑧=𝑝⁰_𝑗⊗𝑝⁰_𝑗⊗𝑝⁰_𝑗(div𝑢)=0. (28)

Theorem 7. The divergence-free scaling function spaces {𝑉_𝑗^div}_{𝑗≥𝑗min} is a multiresolution analysis of ℋ_div(Ω).

Proof. Since ℋ_div(Ω)∩𝑉_𝑗 are a multiresolution analysis of ℋ_div(Ω), it is reduced to prove 𝑉_𝑗^div=ℋ_div(Ω)∩𝑉_𝑗. (29) Noting that (𝑑/𝑑𝑥)𝑉¹_𝑗=𝑉⁰_𝑗 and 𝑉^𝐷_𝑗⊆𝑉¹_𝑗 from (2) and (8), we know 𝑉_𝑗^div⊂𝑉_𝑗. Furthermore, 𝑉_𝑗^div⊂ℋ_div(Ω) by construction. Therefore, 𝑉_𝑗^div⊂ℋ_div(Ω)∩𝑉_𝑗.
Conversely, letting 𝑢∈ℋ_div(Ω)∩𝑉_𝑗, we are going to prove 𝑢∈𝑉_𝑗^div. On the one hand, since 𝑢∈𝑉_𝑗, we have 𝑢=𝑃_𝑗𝑢. On the other hand, since 𝑢∈ℋ_div(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 𝜑=𝑃^𝐷_𝑗𝜑+_{𝑗′≥𝑗}𝑄^𝐷_𝑗′𝜑, (30) where 𝑃^𝐷_𝑗(𝜑)=∑_𝑘𝑐_1,𝑘𝜑⁰_𝑗,𝑘1𝜑^𝐷_𝑗,𝑘2𝜑^𝐷_𝑗,𝑘3𝛿₁+∑_𝑘𝑐_2,𝑘𝜑^𝐷_𝑗,𝑘1𝜑⁰_𝑗,𝑘2𝜑^𝐷_𝑗,𝑘3𝛿₂+∑_𝑘𝑐_3,𝑘𝜑^𝐷_𝑗,𝑘1𝜑^𝐷_𝑗,𝑘2𝜑⁰_𝑗,𝑘3𝛿₃𝑄^𝐷_𝑗′𝜑=_𝑘𝑑_𝑗^1,1_′,𝑘𝜑⁰_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,2_′,𝑘𝜑⁰_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,3_′,𝑘𝜓⁰_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,4_′,𝑘𝜑⁰_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,5_′,𝑘𝜓⁰_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,6_′,𝑘𝜓⁰_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^1,7_′,𝑘𝜓⁰_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}𝛿₁+_𝑘𝑑_𝑗^2,1_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜑⁰_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,2_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜓⁰_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,3_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜑⁰_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,4_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜓⁰_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,5_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜑⁰_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,6_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜓⁰_{𝑗′,𝑘2}𝜑^𝐷_{𝑗′,𝑘3}+𝑑_𝑗^2,7_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜓⁰_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}𝛿₂+_𝑘𝑑_𝑗^3,1_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,2_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜑⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,3_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜑⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,4_′,𝑘𝜑^𝐷_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜓⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,5_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,6_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜑⁰_{𝑗′,𝑘3}+𝑑_𝑗^3,7_′,𝑘𝜓^𝐷_{𝑗′,𝑘1}𝜓^𝐷_{𝑗′,𝑘2}𝜓⁰_{𝑗′,𝑘3}𝛿₃. (31) Since curl[𝜑⁰_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}𝛿₁]∈(𝑉^𝐷_𝑗′⊗𝑉⁰_𝑗′⊗𝑊⁰_𝑗′)×(𝑉⁰_𝑗′⊗𝑉^𝐷_𝑗′⊗𝑊⁰_𝑗′)×(𝑉⁰_𝑗′⊗𝑉⁰_𝑗′⊗𝑊^𝐷_𝑗′), then 𝑃_𝑗curl𝜑⁰_{𝑗′,𝑘1}𝜑^𝐷_{𝑗′,𝑘2}𝜓^𝐷_{𝑗′,𝑘3}𝛿₁=0. (32) Similarly, every term in the right sides of (31) satisfies (32). Finally, 𝑃_𝑗curl𝑄^𝐷_𝑗′𝜑=0. (33) Furthermore, we can obtain 𝑢=𝑃_𝑗curl𝜑=𝑃_𝑗curl𝑃^𝐷_𝑗𝜑=curl𝑃^𝐷_𝑗𝜑. (34) Here, we have used the fact curl(𝑃^𝐷_𝑗𝜑)∈𝑉_𝑗 in the last step of (34). By construction, we have curl(𝑃^𝐷_𝑗𝜑)∈𝑉_𝑗^div, which means 𝑢∈𝑉_𝑗^div and the proof is completed.

Based on the constructive method of vector wavelets and the following decompositions: 𝑉⁰_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗=𝑉⁰_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊⁰_𝑗1⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_2=𝑗min𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑊^𝐷_𝑗3=𝑉⁰_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑉⁰_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_2=𝑗min𝑉⁰_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊⁰_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{2,𝑗3=𝑗min}𝑉⁰_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗3=𝑗min}𝑊⁰_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗2=𝑗min}𝑊⁰_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{1,𝑗2,𝑗3=𝑗min}𝑊⁰_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3,𝑉^𝐷_𝑗⊗𝑉⁰_𝑗⊗𝑉^𝐷_𝑗=𝑉^𝐷_𝑗min⊗𝑉⁰_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑉^𝐷_𝑗min⊗𝑉⁰_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_2=𝑗min𝑉^𝐷_𝑗min⊗𝑊⁰_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊^𝐷_𝑗1⊗𝑉⁰_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{2,𝑗3=𝑗min}𝑉^𝐷_𝑗min⊗𝑊⁰_𝑗2⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗3=𝑗min}𝑊^𝐷_𝑗1⊗𝑉⁰_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗2=𝑗min}𝑊^𝐷_𝑗1⊗𝑊⁰_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{1,𝑗2,𝑗3=𝑗min}𝑊^𝐷_𝑗1⊗𝑊⁰_𝑗2⊗𝑊^𝐷_𝑗3,𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉⁰_𝑗=𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑉⁰_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑊⁰_𝑗3⊕_𝑗^𝑗−1_2=𝑗min𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑉⁰_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑉⁰_𝑗min⊕_𝑗^𝑗−1_{2,𝑗3=𝑗min}𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑊⁰_𝑗3⊕_𝑗^𝑗−1_{1,𝑗3=𝑗min}𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑊⁰_𝑗3⊕_𝑗^𝑗−1_{1,𝑗2=𝑗min}𝑊^𝐷_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑉⁰_𝑗min⊕_𝑗^𝑗−1_{1,𝑗2,𝑗3=𝑗min}𝑊^𝐷_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑊⁰_𝑗3, (35) we can give the definition of anisotropic divergence-free wavelets as follows.

Definition 8. For 𝑗₁, 𝑗₂, and 𝑗₃≥𝑗_min, the anisotropic divergence-free wavelets are defined by Ψ^div,(1,1)_𝐣,𝐤=14^𝑗3𝜑+1curl⁰_{𝑗min,𝑘1}⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₁Ψ^div,(1,2)_𝐣,𝐤=14^𝑗2𝜑+1curl⁰_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₁Ψ^div,(1,3)_𝐣,𝐤=1√2curl𝜓⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₁Ψ^div,(1,4)_𝐣,𝐤=14^𝑗2+4^𝑗3curl𝜑⁰_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₁Ψ^div,(1,5)_𝐣,𝐤=14^𝑗3𝜓+1curl⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₁Ψ^div,(1,6)_𝐣,𝐤=14^𝑗2𝜓+1curl⁰_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₁Ψ^div,(1,7)_𝐣,𝐤=14^𝑗2+4^𝑗3curl𝜓⁰_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₁,Ψ^div,(2,1)_𝐣,𝐤=14^𝑗3𝜑+1curl^𝐷_{𝑗min,𝑘1}⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₂Ψ^div,(2,2)_𝐣,𝐤=1√2curl𝜑^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₂Ψ^div,(2,3)_𝐣,𝐤=14^𝑗1𝜓+1curl^𝐷_𝑗1,𝑘1⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₂Ψ^div,(2,4)_𝐣,𝐤=14^𝑗3𝜑+1curl^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₂Ψ^div,(2,5)_𝐣,𝐤=14^𝑗1+4^𝑗3curl𝜓^𝐷_𝑗1,𝑘1⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₂Ψ^div,(2,6)_𝐣,𝐤=14^𝑗1𝜓+1curl^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₂Ψ^div,(2,7)_𝐣,𝐤=14^𝑗1+4^𝑗3curl𝜓^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₂,Ψ^div,(3,1)_𝐣,𝐤=1√2curl𝜑^𝐷_{𝑗min,𝑘1}⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₃Ψ^div,(3,2)_𝐣,𝐤=14^𝑗2𝜑+1curl^𝐷_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₃Ψ^div,(3,3)_𝐣,𝐤=14^𝑗1𝜓+1curl^𝐷_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜑⁰_{𝑗min,𝑘3}𝛿₃Ψ^div,(3,4)_𝐣,𝐤=14^𝑗2𝜑+1curl^𝐷_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₃Ψ^div,(3,5)_𝐣,𝐤=14^𝑗1𝜓+1curl^𝐷_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₃Ψ^div,(3,6)_𝐣,𝐤=14^𝑗1+4^𝑗2curl𝜓^𝐷_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₃Ψ^div,(3,7)_𝐣,𝐤=14^𝑗1+4^𝑗2curl𝜓^𝐷_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₃. (36)

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 𝑊_𝐣^{div,(𝜀,𝑛)}=spanΨ^{div,(𝜀,𝑛)}_𝐣,𝐤,𝜀=1,2,3,𝑛=1,2,…,7; (37) then 𝑉_𝑗^div=𝑉_𝑗^div_min⊕_{𝑗min≤𝑗1,𝑗2,𝑗3≤𝑗−1}(⊕_{𝜀=1,2,3,𝑛=1,2,…,7}𝑊_𝐣^{div,(𝜀,𝑛)}).

Proof. It can be easily obtained from (35) and Definition 8.

Definition 11. Biorthogonal divergence-free scaling functions and wavelets are defined by Φ^div,1_𝑗,𝐤=1√2𝜑⁰_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝛾_𝑗,𝑘3𝛿₂−𝜑⁰_𝑗,𝑘1⋅𝛾_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3𝛿₃Φ^div,2_𝑗,𝐤=1√2𝛾_𝑗,𝑘1⋅𝜑⁰_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3𝛿₃−𝜑^𝐷_𝑗,𝑘1⋅𝜑⁰_𝑗,𝑘2⋅𝛾_𝑗,𝑘3𝛿₁Φ^div,3_𝑗,𝐤=1√2𝜑^𝐷_𝑗,𝑘1⋅𝛾_𝑗,𝑘2⋅𝜑⁰_𝑗,𝑘3𝛿₁−𝛾_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑⁰_𝑗,𝑘3𝛿₂,Ψ^div,(1,1)_𝐣,𝐤=14^𝑗32+1^𝑗3𝜑⁰_{𝑗min,𝑘1}⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₂−𝜑⁰_{𝑗min,𝑘1}⋅𝛾_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃Ψ^div,(1,2)_𝐣,𝐤=14^𝑗2𝜑+1^{0𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝛾_{𝑗min,𝑘3}𝛿₂−2^𝑗2𝜑⁰_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃Ψ^div,(1,3)_𝐣,𝐤=1√2𝜓⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝛾_{𝑗min,𝑘3}𝛿₂−𝜓⁰_𝑗1,𝑘1⋅𝛾_{𝑗min,𝑘2}⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃Ψ^div,(1,4)_𝐣,𝐤=14^𝑗2+4^𝑗32^𝑗3𝜑⁰_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₂−2^𝑗2𝜑⁰_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃Ψ^div,(1,5)_𝐣,𝐤=14^𝑗32+1^𝑗3𝜓⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₂−𝜓⁰_𝑗1,𝑘1⋅𝛾_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃Ψ^div,(1,6)_𝐣,𝐤=14^𝑗2𝜓+1^0𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝛾_{𝑗min,𝑘3}𝛿₂−2^𝑗2𝜓⁰_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃Ψ^div,(1,7)_𝐣,𝐤=14^𝑗2+4^𝑗32^𝑗3𝜓⁰_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₂−2^𝑗2𝜓⁰_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃,Ψ^div,(2,1)_𝐣,𝐤=14^𝑗3𝛾+1^{𝑗min,𝑘1}⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃−2^𝑗3𝜑^𝐷_{𝑗min,𝑘1}⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₁Ψ^div,(2,2)_𝐣,𝐤=1√2𝛾_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃−𝜑^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝛾_{𝑗min,𝑘3}𝛿₁Ψ^div,(2,3)_𝐣,𝐤=14^𝑗12+1^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃−𝜓^𝐷_𝑗1,𝑘1⋅𝜑⁰_{𝑗min,𝑘2}⋅𝛾_{𝑗min,𝑘3}𝛿₁Ψ^div,(2,4)_𝐣,𝐤=14^𝑗3𝛾+1^{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃−2^𝑗3𝜑^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₁Ψ^div,(2,5)_𝐣,𝐤=14^𝑗1+4^𝑗32^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜑⁰_{𝑗𝑚𝑖𝑛,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃−2^𝑗3𝜓^𝐷_𝑗1,𝑘1⋅𝜑⁰_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₁Ψ^div,(2,6)_𝐣,𝐤=14^𝑗12+1^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}𝛿₃−𝜓^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝛾_{𝑗min,𝑘3}𝛿₁Ψ^div,(2,7)_𝐣,𝐤=14^𝑗1+4^𝑗32^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3𝛿₃−2^𝑗3𝜓^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₁,Ψ^div,(3,1)_𝐣,𝐤=1√2𝜑^𝐷_{𝑗min,𝑘1}⋅𝛾_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₁−𝛾_{𝑗min,𝑘1}⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₂Ψ^div,(3,2)_𝐣,𝐤=14^𝑗22+1^𝑗2𝜑^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₁−𝛾_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₂Ψ^div,(3,3)_𝐣,𝐤=14^𝑗1𝜓+1^{𝐷𝑗1,𝑘1}⋅𝛾_{𝑗min,𝑘2}⋅𝜑⁰_{𝑗min,𝑘3}𝛿₁−2^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜑⁰_{𝑗min,𝑘3}𝛿₂Ψ^div,(3,4)_𝐣,𝐤=14^𝑗22+1^𝑗2𝜑^𝐷_{𝑗min,𝑘1}⋅𝜓⁰_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₁−𝛾_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₂Ψ^div,(3,5)_𝐣,𝐤=14^𝑗1𝜓+1^{𝐷𝑗1,𝑘1}⋅𝛾_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₁−2^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓⁰_𝑗3,𝑘3𝛿₂Ψ^div,(3,6)_𝐣,𝐤=14^𝑗1+4^𝑗22^𝑗2𝜓^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₁−2^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑⁰_{𝑗min,𝑘3}𝛿₂Ψ^div,(3,7)_𝐣,𝐤=14^𝑗1+4^𝑗22^𝑗2𝜓^𝐷_𝑗1,𝑘1⋅𝜓⁰_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₁−2^𝑗1𝜓⁰_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓⁰_𝑗3,𝑘3𝛿₂. (38) Here, 𝛾_𝑗,𝑘=−∫^𝑥₀𝜑^𝐷_𝑗,𝑘(𝑡)𝑑𝑡.

Proposition 12. The families {Φ^div,𝜀_𝑗,𝐤,Ψ^{div,(𝜀,𝑛)}_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗,𝜀=1,2,3,𝑛=1,2,…,7} and {Φ^div,𝜀_𝑗,𝐤,Ψ^{div,(𝜀,𝑛)}_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗,𝜀=1,2,3,𝑛=1,2,…,7} are biorthogonal in (𝐿²(Ω))³.

Proof. It is easily proved by the fact that 𝜓⁰_𝑗,𝑘=−2^𝑗∫^𝑥₀𝜓^𝐷_𝑗,𝑘(𝑡)𝑑𝑡, which is shown in (16).

Theorem 13. The set {Φ_𝑗^div,𝜀_min,𝐤,Ψ^{div,(𝜀,𝑛)}_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗_min,𝜀=1,2,3,𝑛=1,2,…,7} is a Riesz basis of ℋ_div(Ω).

Proof. The completeness is ensured by Theorem 7 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 Ψ^{div,(𝜀,𝑛)}_𝐣,𝐤 are compactly supported, have zero mean value, and belong to the spaces 𝐶^𝜀 for some 𝜀>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=⋃³_𝑘=1Γ_𝑘 with Γ_𝑘=³_{𝑚=1,𝑚≠𝑘}[0,1]^𝑚−1×{0}×[0,1]^3−𝑚,1≤𝑘≤3. (39)

It holds that 𝑢×𝑛=0 on Γ if and only if 𝑢_𝑘=0 on Γ_𝑘(1≤𝑘≤3), which is shown in Figure 3.

Figure 3 

Slip boundary condition (curl).

In this section, we mainly consider the following space: ℋ_curl(Ω)=𝑢∈𝐿²(Ω)³=∶curl𝑢=0,𝑢×𝑛=0on𝜕Ω (40) with free-slip boundary as Figure 4.

Figure 4 

Free-slip boundary condition (curl).

An equivalent characterization is firstly given for ℋ_curl(Ω); and then we will give the MRA and wavelets for it.

Proposition 14. There is the characterization ℋ_curl(Ω)={𝑢=grad𝜑∶𝜑∈𝐻¹₀(Ω)}.

Proof. Suppose 𝜑∈𝐻¹₀(Ω); then 𝑢=grad𝜑=(𝜕₁𝜑,𝜕₂𝜑,𝜕₃𝜑)^𝑇∈(𝐿²(Ω))³. Moreover, =𝜕curl𝑢=curl⋅grad𝜑₂𝜕₃𝜑−𝜕₃𝜕₂𝜑,𝜕₃𝜕₁𝜑−𝜕₁𝜕₃𝜑,𝜕₁𝜕₂𝜑−𝜕₂𝜕₁𝜑^𝑇=0. (41)
Note that 𝜕₁𝜑𝑥,𝑦,0=lim_Δ𝑥→0𝜑𝑥+Δ𝑥,𝑦,0−𝜑𝑥,𝑦,0𝜕Δ𝑥=0,₁𝜑𝑥,𝑦,1=lim_Δ𝑥→0𝜑𝑥+Δ𝑥,𝑦,1−𝜑𝑥,𝑦,1𝜕Δ𝑦=0,₁𝜑(𝑥,0,𝑧)=lim_Δ𝑥→0𝜑(𝑥+Δ𝑥,0,𝑧)−𝜑(𝑥,0,𝑧)𝜕Δ𝑥=0,₁𝜑(𝑥,1,𝑧)=lim_Δ𝑥→0𝜑(𝑥+Δ𝑥,1,𝑧)−𝜑(𝑥,1,𝑧)Δ𝑦=0; (42) therefore, 𝑢₁𝑥,𝑦,0=𝑢₁𝑢𝑥,𝑦,1=0,∀0≤𝑥,𝑦≤1.₁(𝑥,0,𝑧)=𝑢₁(𝑥,1,𝑧)=0,∀0≤𝑥,𝑧≤1. (43) In the same way, one can obtain 𝑢₂𝑥,𝑦,0=𝑢₂𝑢𝑥,𝑦,1=0,∀0≤𝑥,𝑦≤1,₂0,𝑦,𝑧=𝑢₂𝑢1,𝑦,𝑧=0,∀0≤𝑦,𝑧≤1,₃0,𝑦,𝑧=𝑢₃𝑢1,𝑦,𝑧=0,∀0≤𝑦,𝑧≤1,₃(𝑥,0,𝑧)=𝑢₃(𝑥,1,𝑧)=0,∀0≤𝑥,𝑧≤1. (44) This is equivalent to 𝑢×𝑛=0. Therefore, 𝑢=grad𝜑∈ℋ_curl(Ω).
On the other hand, suppose 𝑢∈ℋ_curl(Ω); then we will prove that there exists a function 𝜑∈𝐻¹₀(Ω), such that 𝑢=grad𝜑. Since curl𝑢=0, then 𝜕₂𝑢₃=𝜕₃𝑢₂,𝜕₃𝑢₁=𝜕₁𝑢₃,𝜕₁𝑢₂=𝜕₂𝑢₁. (45) By Stokes formula, there exists a primitive function 𝜑∈𝐻¹(Ω) such that 𝑑𝜑𝑥,𝑦,𝑧=𝑢₁𝑥,𝑦,𝑧𝑑𝑥+𝑢₂𝑥,𝑦,𝑧𝑑𝑦+𝑢₃𝑥,𝑦,𝑧𝑑𝑧. (46) Therefore, 𝜕𝜑/𝜕𝑥=𝑢₁, 𝜕𝜑/𝜕𝑦=𝑢₂, and 𝜕𝜑/𝜕𝑧=𝑢₃; that is 𝑢=grad𝜑. Furthermore, 𝑢×𝑛=0 means that 𝑢₁𝑥,𝑦,0=𝑢₁𝑢𝑥,𝑦,1=0,∀0≤𝑥,𝑦≤1,₂0,𝑦,𝑧=𝑢₂𝑢1,𝑦,𝑧=0,∀0≤𝑦,𝑧≤1,₃(𝑥,0,𝑧)=𝑢₃(𝑥,1,𝑧)=0,∀0≤𝑥,𝑧≤1. (47) Noting that 𝜑𝑥,𝑦,𝑧=^𝑥_𝑥0𝑢₁𝑟,𝑦,𝑧𝑑𝑟=^𝑦_𝑦0𝑢₂()=𝑥,𝑠,𝑧𝑑𝑠_𝑧𝑧0𝑢₃𝑥,𝑦,𝑡𝑑𝑡, (48) 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∈𝐻¹₀(Ω).

Noting that 𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗 is an MRA of (𝐻¹₀(Ω))³, we give the following definition.

Definition 15. For 𝑗≥𝑗_min, the curl-free scaling function spaces 𝑉_𝑗^curl are defined by 𝑉_𝑗^curl=grad𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗=spanΦ^curl_𝑗,𝐤, (49) where the curl-free scaling functions are given by Φ^curl_𝑗,𝐤1=∶√3grad𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3=1√3𝜑^𝐷_𝑗,𝑘1^′⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3,𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2^′⋅𝜑^𝐷_𝑗,𝑘3,𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3^′𝑇. (50)
For convenience, we also consider the standard MRA 𝑉_𝑗 of (𝐿²(Ω))³: 𝑉_𝑗=𝑉⁰_𝑗⊗𝑉¹_𝑗⊗𝑉¹_𝑗×𝑉¹_𝑗⊗𝑉⁰_𝑗⊗𝑉¹₁×𝑉¹_𝑗⊗𝑉¹_𝑗⊗𝑉⁰_𝑗. (51)

Theorem 16. The curl-free scaling function spaces {𝑉_𝑗^curl}_{𝑗≥𝑗min} are a multiresolution analysis of ℋ_curl(Ω).

Proof. Since ℋ_curl(Ω)∩𝑉_𝑗 is a multiresolution analysis of ℋ_curl(Ω), it is reduced to prove 𝑉_𝑗^curl=ℋ_curl(Ω)∩𝑉_𝑗. (52) Noting that (𝑑/𝑑𝑥)𝑉¹_𝑗=𝑉⁰_𝑗 and 𝑉^𝐷_𝑗⊆𝑉¹_𝑗, we know 𝑉_𝑗^curl⊂𝑉_𝑗. Furthermore, 𝑉_𝑗^curl⊂ℋ_curl(Ω) by construction. Therefore, 𝑉_𝑗^curl⊂ℋ_curl(Ω)∩𝑉_𝑗.
Conversely, let 𝑢∈ℋ_curl(Ω)∩𝑉_𝑗, we are going to prove 𝑢∈𝑉_𝑗^curl. 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∈ℋ_curl(Ω), there exists a 𝜑∈𝐻¹₀(Ω) such that 𝑢=grad𝜑. Thus, 𝑢=𝑃_𝑗grad𝜑. (53) Since (𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗)_{𝑗≥𝑗min} forms an MRA of 𝐻¹₀(Ω), we can decompose 𝜑 as 𝜑=𝑃^𝐷_𝑗𝜑+_{𝑗1,𝑗2,𝑗3}⁷_{≥𝑗𝑛=1}𝑄^𝐷_𝑛,𝐽𝜑,𝐽=𝑗₁,𝑗₂,𝑗₃, (54) where 𝑃^𝐷_𝑗𝜑=_𝐤𝑐_𝐤𝜑^𝐷_𝑗,𝑘1𝜑^𝐷_𝑗,𝑘2𝜑^𝐷_𝑗,𝑘3,𝑄^𝐷_1,𝐽𝜑=_{𝑗3≥𝑗𝐤}𝑑¹_𝑗3,𝐤𝜑^𝐷_𝑗,𝑘1𝜑^𝐷_𝑗,𝑘2𝜓^𝐷_𝑗3,𝑘3,𝑄^𝐷_2,𝐽𝜑=_{𝑗2≥𝑗𝐤}𝑑²_𝑗2,𝐤𝜑^𝐷_𝑗,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜑^𝐷_𝑗,𝑘3,𝑄^𝐷_3,𝐽𝜑=_{𝑗1≥𝑗𝐤}𝑑³_𝑗1,𝐤𝜓^𝐷_𝑗1,𝑘1𝜑^𝐷_𝑗,𝑘2𝜑^𝐷_𝑗,𝑘3,𝑄^𝐷_4,𝐽𝜑=_{𝑗2,𝑗3≥𝑗𝐤}𝑑⁴_{𝑗2,𝑗3,𝐤}𝜑^𝐷_𝑗,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3,𝑄^𝐷_5,𝐽𝜑=_{𝑗1,𝑗3≥𝑗𝐤}𝑑⁵_{𝑗1,𝑗3,𝐤}𝜓^𝐷_𝑗1,𝑘1𝜑^𝐷_𝑗,𝑘2𝜓^𝐷_𝑗3,𝑘3,𝑄^𝐷_6,𝐽𝜑=_{𝑗1,𝑗2≥𝑗𝐤}𝑑⁶_{𝑗1,𝑗2,𝐤}𝜓^𝐷_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜑^𝐷_𝑗,𝑘3,𝑄^𝐷_7,𝐽𝜑=_{𝑗1,𝑗2,𝑗3≥𝑗𝐤}𝑑⁷_{𝑗1,𝑗2,𝑗3,𝐤}𝜓^𝐷_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3 (55) are the biorthogonal projectors on, respectively, 𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗, 𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑊^𝐷_𝑗3, 𝑉^𝐷_𝑗⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗, 𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗, 𝑉^𝐷_𝑗⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3, 𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗⊗𝑊^𝐷_𝑗3, 𝑊^𝐷_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗, and 𝑊^𝐷_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3.
Noting that grad𝜑^𝐷_𝑗,𝑘1𝜑^𝐷_𝑗,𝑘2𝜓^𝐷_𝑗3,𝑘3∈𝑉⁰_𝑗⊗𝑉^𝐷_𝑗⊗𝑊^𝐷_𝑗3×𝑉^𝐷_𝑗⊗𝑉⁰_𝑗⊗𝑊^𝐷_𝑗3×𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑊⁰_𝑗3, (56) then 𝑃_𝑗[grad𝑄^𝐷_1,𝐽(𝜑)]=0. Similarly, 𝑃_𝑗[grad𝑄^𝐷_𝑛,𝐽(𝜑)]=0 for 2≤𝑛≤7. Therefore, 𝑢=𝑃_𝑗grad𝜑=𝑃_𝑗grad𝑃^𝐷_𝑗𝜑. (57) Since  grad𝑃^𝐷_𝑗(𝜑)∈𝑉_𝑗^curl⊂𝑉_𝑗, then we obtain 𝑢=grad𝑃^𝐷_𝑗𝜑∈𝑉_𝑗^curl. (58)

Definition 17. For 𝑗₁, 𝑗₂, and 𝑗₃≥𝑗_min, the anisotropic curl-free wavelets and wavelet spaces are defined by Ψ^curl,1_𝐣,𝐤=14^𝑗3𝜑+2grad^𝐷_{𝑗min,𝑘1}⋅𝜑^𝐷_jmin,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3Ψ^curl,2_𝐣,𝐤=14^𝑗2𝜑+2grad^𝐷_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}Ψ^curl,3_𝐣,𝐤=14^𝑗1𝜓+2grad^𝐷_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜑^𝐷_{𝑗min,𝑘3}Ψ^curl,4_𝐣,𝐤=14^𝑗2+4^𝑗3𝜑+1grad^𝐷_{𝑗min,𝑘1}⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3Ψ^curl,5_𝐣,𝐤=14^𝑗1+4^𝑗3𝜓+1grad^𝐷_𝑗1,𝑘1⋅𝜑^𝐷_{𝑗min,𝑘2}⋅𝜓^𝐷_𝑗3,𝑘3Ψ^curl,6_𝐣,𝐤=14^𝑗1+4^𝑗2𝜓+1grad^𝐷_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜑^𝐷_{𝑗min,𝑘3}Ψ^curl,7_𝐣,𝐤=14^𝑗1+4^𝑗2+4^𝑗3grad𝜓^𝐷_𝑗1,𝑘1⋅𝜓^𝐷_𝑗2,𝑘2⋅𝜓^𝐷_𝑗3,𝑘3. (59)

Proposition 18. Defining the wavelet spaces 𝑊_𝐣^curl,𝑛=span{Ψ^curl,𝑛_𝐣,𝐤} for 𝑛=1,2,…,7, then 𝑉_𝑗^curl=𝑉_𝑗^curl_min⊕_{𝑗min≤𝑗1,𝑗2,𝑗3≤𝑗−1}⊕_{𝜀=1,2,…,7}𝑊_𝐣^curl,𝜀. (60)

Proof. The result follows from the following fact: 𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗⊗𝑉^𝐷_𝑗=𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_2=𝑗min𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑊^𝐷_𝑗3=𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_3=𝑗min𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_2=𝑗min𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_1=𝑗min𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{2,𝑗3=𝑗min}𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗3=𝑗min}𝑊^𝐷_𝑗1⊗𝑉^𝐷_𝑗min⊗𝑊^𝐷_𝑗3⊕_𝑗^𝑗−1_{1,𝑗2=𝑗min}𝑊^𝐷_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑉^𝐷_𝑗min⊕_𝑗^𝑗−1_{1,𝑗2,𝑗3=𝑗min}𝑊^D_𝑗1⊗𝑊^𝐷_𝑗2⊗𝑊^𝐷_𝑗3. (61)

Definition 19. Biorthogonal curl-free scaling functions and wavelets are defined by Φ^curl_𝑗,𝐤=1√3𝛾_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3𝛿₁+𝜑^𝐷_𝑗,𝑘1⋅𝛾_𝑗,𝑘2⋅𝜑^𝐷_𝑗,𝑘3𝛿₂+𝜑^𝐷_𝑗,𝑘1⋅𝜑^𝐷_𝑗,𝑘2⋅𝛾_𝑗,𝑘3𝛿₃,Ψ^curl,1_𝐣,𝐤=14^𝑗3𝛾+2^{𝑗min,𝑘1}𝜑^𝐷_{𝑗min,𝑘2}𝜓^𝐷_𝑗3,𝑘3𝛿₁+𝜑^𝐷_{𝑗min,𝑘1}𝛾_{𝑗min,𝑘2}𝜓^𝐷_𝑗3,𝑘3𝛿₂+2^𝑗3𝜑^𝐷_{𝑗min,𝑘1}𝜑^𝐷_{𝑗min,𝑘2}𝜓⁰_𝑗3,𝑘3𝛿₃,Ψ^curl,2_𝐣,𝐤=14^𝑗2𝛾+2^{𝑗min,𝑘1}𝜓^𝐷_𝑗2,𝑘2𝜑^𝐷_{𝑗min,𝑘3}𝛿₁+2^𝑗2𝜑^𝐷_{𝑗min,𝑘1}𝜓⁰_𝑗2,𝑘2𝜑^𝐷_{𝑗min,𝑘3}𝛿₂+𝜑^𝐷_{𝑗min,𝑘1}𝜓^𝐷_𝑗2,𝑘2𝛾_{𝑗min,𝑘3}𝛿₃,Ψ^curl,3_𝐣,𝐤=14^𝑗12+2^𝑗1𝜓⁰_𝑗1,𝑘1𝜑^𝐷_{𝑗min,𝑘2}𝜑^𝐷_{𝑗min,𝑘3}𝛿₁+𝜓^𝐷_𝑗1,𝑘1𝛾_{𝑗min,𝑘2}𝜑^𝐷_{𝑗min,𝑘3}𝛿₂+𝜓^𝐷_𝑗1,𝑘1𝜑^𝐷_{𝑗min,𝑘2}𝛾_{𝑗min,𝑘3}𝛿₃,Ψ^curl,4_𝐣,𝐤=14^𝑗2+4^𝑗3𝛾+1^{𝑗min,𝑘1}𝜓^𝐷_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3𝛿₁+2^𝑗2𝜑^𝐷_{𝑗min,𝑘1}𝜓⁰_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3𝛿₂+2^𝑗3𝜑^𝐷_{𝑗min,𝑘1}𝜓^𝐷_𝑗2,𝑘2𝜓⁰_𝑗3,𝑘3𝛿₃,Ψ^curl,5_𝐣,𝐤=14^𝑗1+4^𝑗32+1^𝑗1𝜓⁰_𝑗1,𝑘1𝜑^𝐷_{𝑗min,𝑘2}𝜓^𝐷_𝑗3,𝑘3𝛿₁+𝜓^𝐷_𝑗1,𝑘1𝛾_{𝑗min,𝑘2}𝜓^𝐷_𝑗3,𝑘3𝛿₂+2^𝑗3𝜓^𝐷_𝑗1,𝑘1𝜑^𝐷_{𝑗min,𝑘2}𝜓⁰_𝑗3,𝑘3𝛿₃,Ψ^curl,6_𝐣,𝐤=14^𝑗1+4^𝑗22+1^𝑗1𝜓⁰_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜑^𝐷_{𝑗min,𝑘3}𝛿₁+2^𝑗2𝜓^𝐷_𝑗1,𝑘1𝜓⁰_𝑗2,𝑘2𝜑^𝐷_{𝑗min,𝑘3}𝛿₂+𝜓^𝐷_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝛾_{𝑗min,𝑘3}𝛿₃,Ψ^curl,7_𝐣,𝐤=14^𝑗1+4^𝑗2+4^𝑗32^𝑗1𝜓⁰_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3𝛿₁+2^𝑗2𝜓^𝐷_𝑗1,𝑘1𝜓⁰_𝑗2,𝑘2𝜓^𝐷_𝑗3,𝑘3𝛿₂+2^𝑗3𝜓^𝐷_𝑗1,𝑘1𝜓^𝐷_𝑗2,𝑘2𝜓⁰_𝑗3,𝑘3𝛿₃. (62) Here, 𝛾_𝑗,𝑘 is defined as in Definition 11.

Proposition 20. The families {Φ^curl_𝑗,𝐤,Ψ^curl,𝜀_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗,𝜀=1,2,…,7} and {Φ^curl_𝑗,𝐤,Ψ^curl,𝜀_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗,𝜀=1,2,…,7} are biorthogonal in (𝐿²(Ω))³.

Theorem 21. The set {Φ_𝑗^curl_min,𝐤,Ψ^curl,𝜀_𝐣,𝐤∶𝑗₁,𝑗₂,𝑗₃≥𝑗_min,𝜀=1,2,…,7} is a Riesz basis of ℋ_curl(Ω).

Proof. It can be proved by the same method as Theorem 13.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (nos. 11201094 and 11161014), Guangxi Natural Science Foundation (no. 2013GXNSFAA019330) and the fund of Education Department of Guangxi (nos. 201012MS094 and 201102ZD015).