The Galerkin Method Based On Daubechies Wavelet

Wavelet Analysis is one of the most popular fields in science research recently in the world which has offered a powerful tool and brought the original ideas to some of the correlative subjects. Wavelet Analysis is a breakthrough progress after Fourier analysis and has caused the extensive concern in science and technology. It is not only include the abundant mathematics theory but also powerful method in the engineering.The definition of Wavelet Analysis is that shifting the mother wavelet e.g. which has limited energy, such as shifting for, then proceeding dimensions, we obtained wavelet basis function ,, which make the inside product to the function to analysis With the Fourier transformation to the and, then the equivalent frequency spectrum of wavelet transformation can be expressed aswhere , are the transformations of , respectively and i is an imaginary number unit. We can give a simple comparison to the wavelet transformation as the Figure below. Using the lens observe target function , represent the function of lens ( for example: the filter or wrapped integral). is equal to make lens opposite in the parallel ambulation in target, equal to make lens opposite to push forward or keep off in the target. Figure explanation of the wavelet transformationThe essence of the Finite Element Method can be summed up to the combination of the Variational Principle and Approximation Space. The traditional finite element method adopted the lower order Band-Emerging interpolation polynomials as the approximation space. However, the generalized finite element method can adopt more extensive basis functions as approximation space. In this paper, the wavelet analysis and the finite element were combined and Daubechies wavelet basis functions replace the interpolation polynomials as the trial function which were applied to the Galerkin method. Generally speaking, this text divide into six sections. The first part gives a general description of the basic thinkings of the finite element method and wavelet analysis. The second part establishes the basic concept of functional , especially three important spaces extensively applied in engineering analysis: Distance space, linear space of assigning norm , the Hilbert space and their quality, then explain the definition, quality of the wavelet analysis and MultiResolution Analysis and Mallat arithmetic which were important in wavelet analysis.The third part explained the wavelet finite element method from the angle of the generalize finite element method. The essence of the Finite Element Method can be summed up to the combination of the Variational Principle and Approximation Space. The definition of the wavelet finite element method is that wavelet basis functions replace the interpolation polynomials as the trial function. Here we deduced the finite element method of the Variational Principle of the boundary of fourth second ellipse equation and the boundary of parabola equation.The fourth part of the paper is using Daubechies wavelets which have the compactly supported set , orthogonality and high order of vanishing moments as the interpolation functions. The differential coefficient of the dimension function indispensable to finite element equations and the Daubechies-Galerkin coefficient are given. The fifth part gave the numerical examples of the application of Daubechies-Galerkin method in the plate bend and parabola equation. Finally, the summary and the prospect of the paper are given.