Guangqing Long

Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations

In this paper we discuss the discrete Legendre Galerkin and discrete Legendre collocation methods for Fredholm-Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both discrete Legendre Galerkin and discrete Legendre collocation solutions in both infinity and L 2 -norm. Numerical examples are given to illustrate the theoretical results.

Legendre multi-projection methods for solving eigenvalue problems for a compact integral operator

In this paper, we consider the M-Galerkin and M-collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels, using Legendre polynomial bases. We obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both Legendre M-Galerkin and Legendre M-collocation methods. We also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both L^2 and infinity norm. We illustrate our results with numerical examples.

The multi-projection method for weakly singular Fredholm integral equations of the second kind

In this paper, we propose a multi-projection method and its re-iterated algorithm for solving weakly singular Fredholm integral equations of the second kind. We apply our methods to Petrov–Galerkin versions to establish excellent superconvergence results, and we illustrate our theoretical results with a numerical example.

Legendre Spectral Projection Methods for Fredholm---Hammerstein Integral Equations

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Hammerstein integral equation. The convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. We are able to obtain similar superconvergence rates for the iterated Legendre Galerkin solution for Hammerstein integral equations with smooth kernel as in the case of piecewise polynomial based Galerkin method.

Discrete multi-projection methods for eigen-problems of compact integral operators

Abstract In this paper, a discrete multi-projection method is developed for solving the eigenvalue problem of a compact integral operator with a smooth kernel. We propose a theoretical framework for analysis of the convergence of these methods. The theory is then applied to establish super-convergence results of the corresponding discrete Galerkin method, collocation method and their iterated solutions. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.

Superconvergence of functional approximation methods for integral equations

In this work, a functional approximation method for calculating the linear functional of the solution of second-kind Fredholm integral equations is developed. When the method is applied to the collocation method or to the multi-projection method, it generates approximations which exhibit superconvergence.

A fast multiscale Kantorovich method for weakly singular integral equations

In this paper, we use the idea of Kantorovich regularization to develop the fast multiscale Kantorovich method and the fast iterated multiscale Kantorovich method. For some kinds of weakly singular integral equations with nonsmooth inhomogeneous terms, we show that our two proposed methods can still obtain the optimal order of convergence and superconvergence order, respectively. Numerical examples are given to demonstrate the efficiency of the methods.

Iterated Fast Collocation Methods for Integral Equations of the Second Kind

In this paper a new iteration technique is proposed based on fast multiscale collocation methods of Chen et al. (SIAM J Numer Anal 40:344–375, 2002) for Fredholm integral equations of the second kind. It is shown that an additional order of convergence is obtained for each iteration even if the exact solution of the integral equation is non-smooth, the kernel of the integral operator is weakly singular and the matrix compression is implemented. When the solution is smooth, this leads to superconvergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.

Discrete Legendre Spectral Galerkin Method for Urysohn Integral Equations

ABSTRACT In this paper, we consider the discrete Legendre spectral Galerkin method to approximate the solution of Urysohn integral equation with smooth kernel. The convergence of the approximate and iterated approximate solutions to the actual solution is discussed and the rates of convergence are obtained. In particular we have shown that, when the quadrature rule is of certain degree of precision, the superconvergence rates for the iterated Legendre spectral Galerkin method are maintained in the discrete case.

Iterated fast multiscale Galerkin methods for eigen-problems of compact integral operators

An iterated fast multiscale Galerkin method is developed for solving the eigen-problem of integral operators with weakly singular kernels. We propose a theoretical framework for analysis of the convergence of these methods and show the fast multiscale Galerkin method obtain the optimal convergence order for eigenvectors and superconvergence order for eigenvalues while the computational complexity for coefficient matrix is almost optimal. The iterated fast multiscale Galerkin method can improve the convergence for eigenvectors and exhibit superconvergence through the iteration technique. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.