Some Results On The Multiscale Numerical Method For Eigen-Problems Of Compact Integral Operators

The main goal of study in this paper is to discuss the numerical method of eigen-problems for compact integral operators,the paper is divided into three chapters,there are two chapters respectively constructed a new numerical method for solving the eigen-problems of compact integral operators,and study the convergence of the numerical method.We use the properties of the multiscal space and the wavelet,and we combine it with the numerical method for linear system and traditional numerical method of eigen-problems with integral operators,then get a more accurate and more efficient numerical method for solving the eigen-problems of integral operators.The full text is divided into four chapter:In chapter 2,mainly is to construct a numerical method for eigen-problems of compact integral operators based on the Gauss-seidel iteration and Jacobi iteration and combining with the multiscal Galerkin method.In the section 1,we introduced the definition and properties of multiscale wavelet space and multiscal wavelet func-tion.In the second section,mainly introduce the framework of the decompositon theory for the eigen-problems.In the third section,according to the framework,define the matrix form of the integral operator then get the coefficient matrix for-m based on the multiscal wavelet space,and we construct the numerical method for eigen-problem with compact integral operators by using the iteration numerical method for linear system,at the end of this chapter,we will present the convergence of these numerical methods for solving eigen-problems.In chapter 3,we committed to construct lultilevel augmentation methods for solving eigen-problems of compact integral operators.In this chapter,firstly we rewrite the eigen-problems of compact integral operators by using multiscal Galerkin method,according to this,we define the matrix form of the integral operator and partition the matrix.Finally,we submit a new numerical method for solving the eigen-problem,at the same time,based on the definition and properties of the wavelet space and wavelet function,we present the matrix form according to the operator form.In order to present the numerical method is reasonable and effective,we proposed a auxiliary numerical method for solving eigen-problems and study the estimation error of approximation solution.While the level of the wavelet space is increased,the calculation of matrix is also increase,but as a result of the matrix block with the multilevel augmentation method is reasonable,the advantage is to simplify the complexity calculation.In chapter 4,we mainly at constructing a fast iteration numerical method for solving eigen-problems with integral operators.In this chapter,similarly with others chapter,we rewrite the eigen-problems and define its operator matrix form then get the equivalent matrix form,this matrix is defense.so in this chapter we introduce a parameter called block parameter,and use the block parameter to block the defense matrix to make it convenience.In the section 3,in order to present the schemes,we introduced integral method based on singularity.Finally,according to the numerical method mention in the section 3,we construct the fast iteration numerical method for solving eigen-problems of integral operators.