The Construction And Application Of The Second Generation Wavelet Based On Lifting Scheme

The construction of the second generation wavelet based on the lifting scheme and the framework for solving partial differential equation method is presented in this paper. This paper also gives a multiresolution finite element method.Firstly, this paper introduces the background and development of the second wavelet, based on which this paper discusses the lifting scheme used for constructing the second generation wavelet, which is also called the second generation transform. It keeps the properties of the first generation wavelet, it doesn't rely on the Fourier transform, which is the main difference. The main principle of lifting scheme is introduced. And this paper also constructs second generation wavelets with one and four vanishing moments.Secondly, this paper develops a numerical method for the transient model in the chemical engineering. It takes advantage of wavelet approximation and the second generation wavelet collocation. Based on the analysis of wavelet coefficients, this algorithm can follow the local structure adaptively, due to the collocation nature of the algorithm, it is easier to deal with nonlinear terms than other wavelet methods. In this paper, making full advantages of the interpolation of the second generation wavelet and the method of interpolate derivatives, this paper gives a interpolate approximation, which greatly deduce the computation. This paper makes use of wavelet collocation for spatial discretization, and then built the ordinary differential equation to time. Then the Runge—kutta method is used to solve the system equation. This paper also gives the analysis of the convergence.Finally, based on the second generation wavelet, a multiresolution finite element method is presented, and the corresponding adaptive algorithm is constructed. The hierarchical approximation spaces for finite element analysis are produced and the decoupling condition of the stiffness matrix of the finite element equation is established by introducing wavelet vanishing moments. A class of second generation wavelet-based beam elements is constructed by using second generation wavelet function. It is realized that the complicated beams such as those with unequal cross section, local load and so on, can be analyzed by this element. The numerical examples illustrate that the second generation wavelet-based elements has high analytical accuracy for beam bending problems with its various boundary conditions...