Anisotropic curl-free wavelets with boundary conditions
© Jiang; licensee Springer 2012
Received: 3 March 2012
Accepted: 6 September 2012
Published: 19 September 2012
This paper deals with the construction of anisotropic curl-free wavelets that satisfy the tangent boundary conditions on bounded domains. Based on some assumptions, we first obtain the desired curl-free Riesz wavelet bases through the orthogonal decomposition of vector-valued . Next, the characterization of Sobolev spaces is studied. Finally, we give the concrete construction of wavelets satisfying the initial assumptions.
Due to their potential use in many physical problems, like the simulation of incompressible fluids or electromagnetism, curl-free wavelet bases have been advocated in several papers and all results focus on the cases of and [1–4]. Moreover, it is questionable whether they are appropriately called bases and whether they can be used to characterize Sobolev spaces. However, it is reasonable to study the corresponding wavelet bases on bounded domains because of some practical use. At the same time, the boundary conditions, the stability and the characterization of Sobolev spaces are also necessary in some applications such as adaptive wavelet methods. In references [5, 6], anisotropic divergence-free wavelets which satisfy the specific boundary conditions on the hypercube are studied. Inspired by the fact that a div-free space and a curl-free space form the orthogonal Helmholtz decomposition, we mainly study the anisotropic curl-free wavelet bases satisfying the tangent boundary conditions on bounded domains in this paper, which is organized as follows. In Section 2, based on some assumption, the desired curl-free wavelets are constructed through the orthogonal decomposition of vector-valued . Section 3 is devoted to studying the characterization of Sobolev spaces. We give the concrete construction of wavelets satisfying the initial assumption in the final section.
2 Decomposition of
In this part, we will construct curl-free wavelets that satisfy tangent boundary conditions by the orthogonal decomposition of vector-valued .
Finally, let , .
The following result will be proved in Section 4:
Proposition 2.1 It holds that , and are Riesz bases for (), and , respectively.
the remaining results can be proved similarly. □
for any , then . On the other hand, since , then . Therefore, . □
Finally, we obtain and .
we obtain and .
In view of Proposition 2.1, we obtain
Theorem 2.1 In the situation of Assumption 2.1, the collections () are Riesz bases for (). and are Riesz bases for and , respectively.
Note 2.1 In fact, for . , which is defined in Section 4.
3 Characterization of
The following result will be verified in Section 4:
Based on this assumption, we obtain:
Theorem 3.1 In the situation of Assumptions 2.1 and 3.1, the collection , normalized in , is a Riesz basis for .
with . □
4 Construction of wavelets
In this section, we will give the construction of wavelets satisfying Assumptions 2.1 and 3.1.
Lemma 4.1 ([, Corollary 3.3])
Moreover, and are bi-orthogonal.
Note 4.1 It has been pointed out in  that such wavelet bases can be obtained by taking standard bi-orthogonal wavelet bases for that satisfy the corresponding Jackson and Bernstein assumptions of d, , γ and with and reading as and (see ), and then removing those scaling functions without a vanishing moment.
The following result can be proved by the same method as Corollary 3.7 of .
respectively. For , the corresponding collections are bi-orthogonal in .
From Corollary 4.1 and the definition of , we obtain
and , respectively. For , the collections are bi-orthogonal Riesz bases for .
Applying the property of an orthogonal transform, we infer the following result.
and , respectively. In particular, for , the collections and are bi-orthogonal Riesz bases for .
In the following, we are mainly concerned with the cases and because of the complicated form of curl operators in .
(), or .
is a Riesz basis for the vector valued space .
- (ii)Since , taking in Proposition 4.2, we know the set is a Riesz basis for . Furthermore, it is easy to verify
The project is supported by the National Natural Science Foundation of China (No. 11201094, 11161014), the 863 Project of China (No. 2012AA011005), the project of Guangxi Innovative Team (No. 2012jjGAG0001), the fund of Education Department of Guangxi Province (No. 201012M9094, 201102ZD015, 201106LX172).
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