Tight Integral Operator Eigenvalue Problem Of Numerical Algorithms

This paper focus on the superconvergence of numerical solutions of eigen-problemsof compact linear integral operators,and fast Petrov-Galerkin algorithms with optimalconvergence for eigen-problem of a compact linear integral operator with a weakly singularkernel. The paper is organized as follows:In chapter 2, Multi-projection method is proposed for eigen-problem of compactintegral operator with a weakly singular kernel. The abbreviation of Multi-projection isM-projection. The M-projection method with superconvergence is for solving the integralequation of second kind. For solving the eigen-problem,we set up a theoretical framworkby M-projection. Then we apply this framwork to the eigen-problem with a weaklysingular kernel to obtain super-convergence results.In chapter 3,a discrete Multi-projection method is developed for sloving the eigen-value a problem of a compact integral operator with a smooth kernel. We propose a theo-retical framwork which is convenient for the analysis of discrete M-projection method Theframwork is then applied to established super-convergence results of the correspondingdiscrete Galerkin method and the collocation method. Numerical examples are preseatedto illustrated the error of these methods.In chapter 4,we apply the smei-biorthogonal multiwavelets [36] to eigenvalue prob-lem with a weakly singular kernel. With this basis．we develop the fast multiscale Petrov-Galerkin algortihms for eigen-problem. The convergence rate and computational complex-ity of the algorithm are analyzed. On the basis of keeping the compuational complexity,we proof our methods obtain the optimal convergence rate with giving the range of thetruncation parameters. Finally,a numerical example is given to testify the theoreticalresult.