The Research Of Wavelet Boundary Element Method For Numerical Solution Of Laplace Equation

In recent years, wavelet boundary element method has developed a new numerical method. Although there is few works about this method, it still shows unique advantages and strong vitality at the beginning. To overcome the shortcoming of singular integral difficulty existed in the natural boundary element method, in this paper, we mainly study the application of the wavelet base function in the natural boundary element method, it not only simplify the calculation process, but also improve the computation accuracy. The basic idea is transforming the differential equations into its equivalent variational problem by the natural boundary naturalization, and then discreting it by the Galerkin-wavelet method or wavelet interpolation to get the corresponding stiffness matrix with unique advantage, it substantially reducing computation. Meanwhile we make the further analysis to the error estimate of the numerical solution. The given example indicates the validity of each kind of wavelet boundary element method.This paper includes five chapters. Chapter 1 introduces the boundary element method, the natural boundary naturalization theory and the wavelet analysis theory. The natural boundary integral equation of Laplace equation and the Possion integral formula by the natural boundary naturalization on the typical domain are also given.In chapter 2, since there is difficulty in solving the Neumann boundary value problem of the Laplace equation on half-plane encounters strongly singular integral. In order to simplify the problem and get more precise numerical solution, we take full advantages of the Shannon wavelet limited bandwidth in frequency domain and use Shannon wavelet boundary element method to solve the problem.In chapter 3, combining the properties of smooth, fast weaken and the interpolation of the wavelet base with the natural boundary element method, it makes the computation reduce and improves accuracy. And it effectively resolves the trouble of the singular integral when solving Laplace equation.In chapter 4, we apply Hermite Cubic Spline Multi-wavelet Natural Boundary Element Method to the Neumann boundary value problem of the Laplace equation on the upper half-plane and make further research. Moreover, the given example proves that the method is effective and flexible.In chapter 5, we applies the criterion of Quak Triangular Wavelet as the base function to discrete Natural Boundary integral equation, and we get the stiffness matrix coefficient formula expressed by one item or two items, which can simply the problem and get more precise solution.