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PDE remains an open topic of active research, and the use of wavelet bases for the solution of PDEs remains a research topic that is currently centered in the mathematics community. Kim[16] and Chiavassa [17] discussed wavelet Galerkin methods for different types of PDEs. Venini[18] employed linear interpolation wavelet to present an adaptive wavelet Galerkin method for analyzing one-dimensional structural problems in the presence of elastic–plastic-damage behaviors. In order to settle the ill-posed problem of the first kind discrete Fred-holm integral equations, Sánchez-ávila[19] presented an adaptive wavelet-based numerical method to detect discontinuities by estimation of its local Hölder exponents and obtain a regularized solution of the original equation. Fang [20] proposed a numerical implementation of the fast Galerkin method for Fredholm integral equations using the piecewise polynomial wavelets. One advan-tage of using the piecewise polynomial wavelet is that it has a close form which provides convenience for numerical computation. The segmentation method with the aid of Haar wavelets was proposed by Lepik [21] , the advantages of the wavelet method were empha-sized by the author, such as detecting singularities, suiting for local high gradient, irregular structure and transient phenomena exhib-ited by the analyzed function. Mehraeen [22] has constructed the linear orthogonal scaling function and wavelet, which are used as shape functions in the multiscale wavelet Galerkin homogeniza-tion. Amaratunga and Sudarshan [23,24] presented an adaptive
multiscale analysis based on operator-customized wavelets for PDEs. Within these whole domain discretization methods, wave-lets defined on the whole square integrable real space L2 were commonly used. To deal with the boundary conditions on a bounded interval, the wavelet basis have to be truncated, which is a difficult task. In addition, the whole domain discretization methods usually fail to accommodate the complex solving domains.
  Chui and Quak [25] have constructed BSWI. In the present work, two-dimensional tensor product BSWI are chosen to serve as the multiscale approximation basis for the construction of multiscale BSWI elements. On a bounded interval, BSWI basis not only has the good characteristics of short support, smoothness, symmetry,
etc. but also has explicit expression [26,27]. Aiming at the nesting approximation of BSWI, the 2D C0 type multiscale BSWI elements are derived in detail. The element displacement field represented by the coefficients of wavelets expansion in wavelet space is trans-formed into the physical degree of freedoms (DOFs) in finite element space via the corresponding transformation matrix.
  Therefore, multiscale BSWI elements can be directly inserted into the mature framework of adaptive finite element analysis with posterior error estimation[28,29]. The numerical examples testify the multiscale BSWI elements can be employed as the adaptive fi-nite element analysis for singularity problems in engineering.

2. B-spline scaling functions and wavelets on the interval

  In this section, a brief description of B-spline scaling functions and wavelets on the interval are given.

References
[1] Cohen A. Numerical analysis of wavelet method. Amsterdam: Elsevier; 2003.
[2] Dahmen W. Wavelet methods for PDEs—some recent developments. J Compu本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。