Some Results On The Numerical Method For Eigenvalues Of Compact Integral Operators

In this paper we develop three numerical methods for solving eigen-problem of compact integral operators.We describe the eigen-problem of compact integral operators and implement the two algorithm in two chapters.Moreover,we setup a new numerical method for solving eigen-problem of integral operators and discuss its convergence.We setup the theoretical framework of the three numerical method under the properties of wavelet functions in multiscale Galerkin space.When the Galerkin method is combined with these methods in this paper,we know that under the same condition,the numerical method in this paper exhibits to be convenient for implemementing adaptivity,and reduces the computational cost greatly and leads to these methods faster.This paper consists of four chapters:In chapter 1,we mainly describes the meaning of using background,main works,basic knowledges and some symbols of this article.In chapter 2,we update the numerical method of Gauss Seidel and Jacobi iteration by using multiscale Galerkin method.Then the convergence of these methods can be fully proved by numerical examples.The second section introduces the theoretical framework of fast multiscale iteration.The third section introduces the fast multiscale Galerkin method.In chapter 3,we mainly describe the implementation of multilevel augmentation algorithm by using the existing theoretical knowledge.In the second section,we mainly describe the multiscale Galerkin method for eigen-problem.The first step of the third section mainly describes the implementation of the wavelet function,second step describes the implementation of double integral,and third step uses power method to solve eigenvalue and eigenvector of compact integral operators.The fourth step describes the implementation of the multilevel augmentation algorithm,which includes the block of compact integral operator matrix,and gives the matrix form of the eigenvalue problem.In chapter 4,we mainly construct a meshless method for eigen-problems of integral operators.In the first section,we describe the theoretical knowledge of MLS approximation and the construction algorithm of basis functions.Therefore,if the parameters of the weight function are far away from each other,the weight function w(x,y)=0,which shows that the weight function has the advantage of greatly simplifying the calculation.In the second section,we mainly describe some basic knowledge of eigen-problems.In section 3,we construct a new numerical method for eigen-problems of compact integral operators,i.e.meshless collocation method.