Research Article | Open Access
Yingchun Jiang, Qingqing Sun, "Three-Dimensional Biorthogonal Divergence-Free and Curl-Free Wavelets with Free-Slip Boundary", Journal of Applied Mathematics, vol. 2013, Article ID 954717, 10 pages, 2013. https://doi.org/10.1155/2013/954717
Three-Dimensional Biorthogonal Divergence-Free and Curl-Free Wavelets with Free-Slip Boundary
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.
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  for divergence-free wavelets in two-dimensional case . 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
In this part, we will introduce the interval wavelets used in the subsequent construction.
The existence of divergence-free and curl-free wavelets on follows from the following fundamental proposition .
Proposition 1. Let be a biorthogonal MRA of , with compactly supported scaling functions and wavelets , such that for . Then there exists a biorthogonal MRA , with associated scaling functions and wavelets , such that The dual functions verify and .
Based on the above proposition, Jouini and Lemarié-Rieusset  proved the existence of two one dimensional MRAs of linked by In the following, we simply introduce the construction of these spaces. Suppose that in Proposition 1 is supported on and reproduces polynomials up to degree : with . Similarly, is supported on and reproduces polynomials up to degree .
For being sufficiently large, the spaces have the structure where is the space whose supports are included into ( be two fixed parameters), and , . Moreover, are the edge scaling functions at the edge 0 being defined by At the edge 1, are defined by symmetry using .
Similarly, the biorthogonal spaces are defined with the same structure as Adjusting the parameters such that The last step of the construction is the biorthogonalization process, since the edge scaling functions of and are no more biorthogonal. Finally, form a biorthogonal MRA of .
As described in , removing the edge scaling functions and leads to Similarly, define . After a biorthogonalization process, we finally note that and the spaces form a biorthogonal MRA of .
The construction of follows the same structure. Since , has compact support and reproduces polynomials up to degree : with . The scaling function has support and reproduces polynomials up to degree . Consider where and supports are included into . The left edge scaling functions are Biorthogonal spaces are similarly defined, but by satisfying vanishing boundary conditions at 0 and 1, then with and for . It is easy to know .
In practice, we choose with 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 can be seen from . Moreover, they satisfy the following result.
Proposition 2 (see ). Let and be MRAs satisfying and ; then the wavelet spaces and are linked to the biorthogonal wavelet spaces associated to by Moreover, let and be two biorthogonal wavelet bases of and . Biorthogonal wavelet bases of and are directly defined by
3. Divergence-Free Wavelets on
In this section, we mainly consider the following space with free-slip boundary as Figure 2
For , the 3D curl-operator is defined as
Remark 3. Taking Fourier transform on the both sides of leads to the equation In , define the following functions Then, according to (20), it is easy to verify that which is equivalent to . Therefore, any function which satisfies can be characterized by curl operator as with . In fact, a similar result holds in 3D nonsmooth domains.
Proposition 4 (see ). There is a characterization
Based on Proposition 4, we give the following definition of divergence-free scaling function spaces.
Definition 5. For , the divergence-free scaling function spaces are defined by
where the divergence-free scaling functions are given by
For proving the consequent main result, we also consider the standard MRA of :
The following conclusion shows that the space preserves the divergence-free condition.
Proposition 6. If and , then , where is the biorthogonal projector on .
Proof. Let ; then Therefore, by the fact in , we obtain
Theorem 7. The divergence-free scaling function spaces is a multiresolution analysis of .
Proof. Since are a multiresolution analysis of , it is reduced to prove
Noting that and from (2) and (8), we know . Furthermore, by construction. Therefore, .
Conversely, letting , we are going to prove . On the one hand, since , we have . On the other hand, since , there exists a such that . Moreover, . Thus, . Furthermore, we can decompose by isotropic vector wavelets as where Since , then Similarly, every term in the right sides of (31) satisfies (32). Finally, Furthermore, we can obtain Here, we have used the fact in the last step of (34). By construction, we have , which means and the proof is completed.
Based on the constructive method of vector wavelets and the following decompositions: we can give the definition of anisotropic divergence-free wavelets as follows.
Definition 8. For , , and , the anisotropic divergence-free wavelets are defined by
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 then .
Definition 11. Biorthogonal divergence-free scaling functions and wavelets are defined by Here, .
Proposition 12. The families and are biorthogonal in .
Proof. It is easily proved by the fact that , which is shown in (16).
Theorem 13. The set is a Riesz basis of .
Proof. The completeness is ensured by Theorem 7 and Proposition 10. Now, it remains to prove the -stability of the basis. By assumption of 1D scaling and wavelet functions, the divergence-free wavelets are compactly supported, have zero mean value, and belong to the spaces for some ; then they constitute a vaguelette-family (). Furthermore, the Riesz stability follows from the existence of a biorthogonal wavelet family given by Proposition 12.
4. Curl-Free Wavelets on
The boundary condition considered in  is on with
It holds that on if and only if on , which is shown in Figure 3.
In this section, we mainly consider the following space: with free-slip boundary as Figure 4.
An equivalent characterization is firstly given for ; and then we will give the MRA and wavelets for it.
Proposition 14. There is the characterization .
Proof. Suppose ; then