Algorithms issued from the wavelet numerical method have been used in many situations and with different discretizations for the resolution of some mathematical and physical partial differ-ential equations (PDEs). 

 基于小波数值方法的算法已经成功地应用于很多领域，采用了不同离散方式去求解一些数学和物理离散方程。

  The wavelet numerical methods embody two prominent advantages. The one is that the scale is di-rectly upgraded by using the so called two scale equations, namely, the scaling functions at different scale are employed directly to form the multiscale approximation basis.   http://www.51lunwen.org/uk/ The other is that the nesting approximation is performed by using the lifting relation-ship between scale and wavelet spaces, i.e. the scaling functions and wavelets at certain scale are adopted to form the scaling func-tion at next scale. Therefore, wavelet numerical methods are well argued by many researchers not only in numerical analysis do-mains but also in structural analysis fields.
  In accordance with the first prominent advantages of wavelet numerical methods, recently, two-dimensional Daubechies wave-let-based element for a thin plate-bending problem had been con-structed by Chen [5]. For Daubechies wavelet lacking of explicit expressions, http://www.51lunwen.org/  the calculation of integral has some difficulties.  Han  proposed multivariable wavelet-based finite element method using C0 type interpolating wavelet to analyze the thick plate. Also, Han constructed some spline wavelet elements for analyzing structural mechanics problems. Those elements were built up in the theory of spline wavelet. Xiang successfully
constructed one-dimensional wavelet-based elements and two-dimensional plane elastomechanics and Mindlin plate elements  by using B-spline wavelet on the interval (BSWI). Moreover, Xiang  employed BSWI scaling functions as approximation functions for solving thin plate bending and vibration problems with good
performances.
  For the nesting approximation of wavelet numerical methods, in the current literatures, the whole domain discretization method was widely used. Martin and Michel made a full investigation about convergence of an adaptive semi-Lagrangian scheme for the Vlasov–Poisson system. Wavelet-based adaptive Galerkin
discretization method was verified to be a robust way to solve elliptic PDE’s on product domains [13] . Dijkema, Christoph and Rob also proposed an adaptive wavelet method for solving high-dimensional elliptic PDEs, such as Poisson’s equation and the wavelet-based approximations converge in energy norm with the same rate as the best approximations form the span of the best N tensor product wavelets[14] . 

  The numerical performance of wave-let methods for PDEs was evaluated by Christon [15] . Some comments were also given by the author, i.e. the design of wavelet bases that are customized for a specific 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[ presented an adaptive wavelet-based numerical method to detect discontinuities by estimation of its local H?lder exponents and obtain a regularized
solution of the original equation.