C～1 Spline Wavelet Methods And Their Applications

Many physical phenomena in nature and many problems in engineering application can be described by partial differential equations. The general partial differential equations have no exact solutions, therefore it is significant to find the numerical solutions for the equations. The traditional solving methods of partial differential equations are mainly the finite difference method, the finite element method and spectral method. The first two methods are flexible for the irregular region, but the computational cost is large and the accuracy is not high. And the spectral method has very high accuracy for the linear and regular region, but it's poor flexible. Recently, the wavelet analysis theory becomes one of the important branches of mathematics, the wavelet possess some properties, such as compact support, vanish moment etc., it has special meaning for the solving of differential equations, therefore people apply various wavelets to solve partial differential equations.In the present thesis, we combine spline wavelet with the finite element method to solve the boundary value problems of partial differential equations. Our spline wavelet functions are spline wavelet functions constructed on Powell-Sabin triangulations, these wavelets have compact support, C1C 1? regularity and H 2? stability. Combining spline wavelet with the finite element method, our method not only keeps the flexibility of the finite element method, but also raises the accuracy of the solutions, accelerates the convergence speed of the solutions and reduces the computational cost. So, our method has applied value broadly. The paper mainly discusses the following contents:1. Introducing traditional methods of partial differential equations and the application of the wavelet analysis in fin- ding the numerical solutions to partial differential equations.2. Presenting the related knowledge of the Powell-Sabin elements, Hermite subdivision scheme and multi-resolution analysis. This lays basis for constructing the wavelet basis and the solution to partial differential equations in this paper.3. Constructing C 1 spline wavelet which have compact support, C 1?regularity and H 2-stability, gives explicit expressions for the wavelets on the three-direction mesh.4. Using spline wavelet method to solve Neumann boundary value problem of partial differential equations, gives the convergence of the solution and estimates: energy norm estimate, norm estimate, C1L2 H ? q( )norm estimate. Finally, a numerical example is presented. q >0We provide a new way to solve differential equations by using spline wavelet on Powell-Sabin triangulations.