Superconvergence of the iterated degenerate kernel method

Abstract

First, in this paper, a general theory for the iterated operator approximation is developed.Some of the known results of the superconvergence of the various iterated schemes can be formulated as special cases of this theory. The method is then subsequently used to prove the superconvergence of the iterated degenerate kernel method for the Fredholm equations of the second kind. A similar result of the superconvergence of the degenerate kernel method for the Hammerstein equations is also given.

Superconvergence of iterated numerical solutions using wavelets

“…More on this point can be found in Ref. 9. However, the sparsity of the resulting matrix is normally lost without the assumption of wavelets with vanishing moments.…”

A degenerate kernel method for eigenvalue problems of compact integral operators

“…By iterating the eigenfunctions, we obtain the similar superconvergence rates for the eigenfunctions as in the iterated Galerkin and iterated collocation methods. The results in this paper depend heavily on the results in [8].…”

“…In [8] a degenerate kernel method is proposed to obtain the superconvergence for the numerical solutions of integral equations of the second kind for a smooth kernel. In this paper we extend their results to the corresponding eigenvalue problem.…”

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

“…In ( Kumar (1987), Kumar and Sloan (1987)), a new type of collocation method was discussed in certain piecewise-polynomial space to establish superconvergence results for single Hammerstein integral equation with smooth kernels, whereas in Kaneko et al (1992), Kaneko et al extended those results for weakly singular kernels. In Kaneko and Xu (1991), a degenerate kernel method was used for the single nonlinear Fredholm-Hammerstein integral equation by approximating the kernel, and it was shown that it converges with order O(h r ), which was improved by iterated degenerate kernel method in Kaneko et al (2001) by obtaining the superconvergence rate as O(h 2r ), where h is the norm of the partition and r is the degree of the piecewise polynomials employed in approximating the kernel. In Kaneko and Xu (1996) and Kaneko et al (1997), Kaneko et al discussed iterated Galerkin and iterated collocation methods, respectively, to obtain the superconvergence results for solving the single Fredholm-Hammerstein integral equation for both smooth and weakly singular kernels.…”