Some Research On Fast Algorithms Of Steklov Eigenvalue Problems

In this paper,we mainly study the wavelet collocation method,wavelet Galerkin method and Fourier Galerkin numerical method for Steklov eigenvalue problem with smooth boundary.Firstly,we use the potential theory to transform the differential Steklov eigenvalue problem into the boundary integral eigenvalue equation.In the framework of spectral projection approximation,we use three numerical methods to solve the boundary integral eigenvalue problem.Finally,we analyze and compare the convergence and computational complexity of the methods,and conclude that the Fourier Galerkin numerical method has the best convergence effect and the least computational complexity.This paper is divided into four chapters: In Chapter 1,we mainly introduce the research background and current situation of Steklov eigenvalue problem,as well as the related research status of multi-scale fast wavelet Galerkin method,multi-scale fast wavelet collocation method,fast wavelet collocation method and fast Fourier Galerkin method.We also describe the abstract framework of the spectral approximation of the eigenvalue problem and the related convergence theory,and use the direct boundary method to convert the Steklov eigenvalue problem into an integral eigenvalue equation,reducing the two-dimensional Steklov eigenvalue by one dimension.In Chapter 2,we mainly study the multi-scale fast wavelet-collocation method for the Steklov eigenvalue problem.First,we use the multi-scale wavelet basis and collocation function to construct a multi-scale space,and then describe the approximate framework of the Steklov eigenvalue problem.The dense matrix is compressed to get the fast multi-scale wavelet collocation method.Next,we conduct the convergence analysis of the fast multi-scale method to get the best convergence.Finally,numerical experiments are given to verify the correctness of the theoretical convergence.Secondly,we compress the dense matrix with the corresponding truncation strategy of wavelet collocation method to obtain the fast multi-scale wavelet collocation method.Then we analyze the convergence of the fast multi-scale method to obtain the optimal convergence.Finally,we give numerical experiments to verify the theoretical convergence.In Chapter 3,we mainly study the multi-scale fast wavelet-Galerkin of the Steklov eigenvalue problem.Similar to the second chapter,we use the multi-scale wavelet basis to construct a multi-scale space to project the approximate solution of the problem,and also adopt the same matrix compression strategy.We also give an estimate of the convergence of the method and the results of an example to prove the convergence of the method.In Chapter 4,we construct the Fourier Galerkin method to solve the Steklov eigenvalue problem by using the Fourier basis.In the first section,we describe the advantages of choosing the Fourier basis to simplify the weak singular kernel integral operator;In the second section,we describe the approximate theoretical framework of Fourier Galerkin method;In the third section,we use the truncation strategy to compress the dense matrix corresponding to the smooth kernel,and estimate the number of non-zero terms of the compressed matrix to obtain the fast Fourier Galerkin method;In the fourth section,we analyze the convergence of the fast Fourier Galerkin method and obtain the optimal convergence.In the last section,we verify the convergence and computational complexity of the fast Fourier Galerkin method by numerical examples.