Superconvergence And Fast Methods For Fredholm Integral Equations Of The Second Kind

This Phd thesis mainly focus on the superconvergence of numerical solutions of integral equations of second kind, and fast Petrov-Galekrin algorithms with optimal convergence for singular integral equations of second kind. The thesis is organized as follows:In Chapter 2, discrete multi-projection method is proposed for second kind integral equations, we set up a theoretical framework which is convenient for the analysis of discrete M-projection methods and corresponding iterated solutions. We apply this theory to discrete M-Galerkin methods and discrete M-collocation methods to obtain superconvergence theorems.In chapter 3, we propose the asymptotic error expansions of iterative solutions of discrete multi-projection methods and its Richardson extrapolations, we show that provided the solution is sufficiently smooth, the iterated solution of M-projection methods admit an error expansion in the powers of the step-size h, beginning with terms in h^4k, where k is the order of piecewise polynomials. Thus Richardson's extrapolation can be performed based on this error expansion, this will increase the accuracy of the numerical solution greatly.In chapter 4, we propose and analyze two iteration methods %r Fredholm integral equations of second kind. we show that for every step of iteration the coefficient matrix remains same as the original approximation method, while we can obtain the superconvergence rates for every step of iteration. Our methods are suitable to not only orthogonal projection methods but interpolation projection methods. We apply our methods to M-projection methods, Galerkin methods, collocation methods and degenerated kernel methods, respectively.In chapter 5, we purpose the fast Petrov-Galerkin methods with optimal order for singular integral equations of second kind. We construct two wavelet basis with having semi-biorthogonal property, small support and higher order vanishing moment. With this basis, we develop the theoretical framework of fast Petrov-Galerkin methods for singular integral equations, and prove our methods obtain optimal order of convergence and almost optimal complexity.