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Journal of Applied Mathematics
Volume 2013, Article ID 954717, 10 pages
http://dx.doi.org/10.1155/2013/954717
Research Article

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

School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin 541004, China

Received 17 August 2013; Accepted 16 November 2013

Academic Editor: Antonio J. M. Ferreira

Copyright © 2013 Yingchun Jiang and Qingqing Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [15]. 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

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 [11].

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 [12] 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 [13], 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 [13]. Moreover, they satisfy the following result.

Proposition 2 (see [12]). 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

Let and let be the normal vector; the boundary condition considered in [6] is on with It holds that on if and only if on . We call it a slip boundary, which is shown in Figure 1.

954717.fig.001
Figure 1: Slip boundary condition (divergence).

In this section, we mainly consider the following space with free-slip boundary as Figure 2

954717.fig.002
Figure 2: Free-slip boundary condition (divergence).

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 [14]). 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 [10], 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 .

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

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 ([12]). 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 [7] is on with

It holds that on if and only if on , which is shown in Figure 3.

954717.fig.003
Figure 3: Slip boundary condition (curl).

In this section, we mainly consider the following space: with free-slip boundary as Figure 4.

954717.fig.004
Figure 4: Free-slip boundary condition (curl).

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 . Moreover,
Note that therefore, In the same way, one can obtain This is equivalent to . Therefore, .
On the other hand, suppose ; then we will prove that there exists a function , such that . Since , then By Stokes formula, there exists a primitive function such that Therefore, , , and ; that is . Furthermore, means that Noting that we obtain , , and . Therefore, .

Noting that is an MRA of , we give the following definition.

Definition 15. For , the curl-free scaling function spaces are defined by where the curl-free scaling functions are given by
For convenience, we also consider the standard MRA of :

Theorem 16. The curl-free scaling function spaces are a multiresolution analysis of .

Proof. Since is a multiresolution analysis of , it is reduced to prove Noting that