Superconvergence of the iterated collocation methods for Hammerstein equations

Abstract

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [ 161 concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 181.

Superconvergence of iterated numerical solutions using wavelets

“…In the case of Fredholm integral equations of the second kind with weakly singular kernels, a certain enhancement in the convergence rate was also obtained for the iterated variants 3 . We also remark that many results on the superconvergence of the iterates for the Fredholm equations have been generalized to a class of nonlinear Hammerstein equations 2,3 . In this paper, we obtain a superconvergence of the iterated degenerate kernel method using the wavelets of Alpert.…”

“…With a sufficiently smooth kernel 1,10 , the rate of convergence of the iterated variants is twice as fast as that of the original convergence, a phenomenon commonly known as superconvergence. We point out that it is somewhat more involved to get a superconvergence of the iterated collocation variants than that of the iterated Galerkin method 1,2 . In the case of Fredholm integral equations of the second kind with weakly singular kernels, a certain enhancement in the convergence rate was also obtained for the iterated variants 3 .…”

“…The use of wavelets in the degenerate kernel method plays a critical role in obtaining this superconvergence which is the convergence that converges faster than generally expected. Superconvergence of the iterated variants for the Galerkin method as well as the collocation method has been studied extensively in recent years [1][2][3] . We remark that, as the method of Alpert is the Galerkin method, we expect that there would be a superconvergence when this solution is iterated.…”

Superconvergence of interpolated collocation solutions for Hammerstein equations

“…The superconvergence of the iterated Galerkin method for Hammerstein equations with smooth as well as weakly singular kernels was probed by Kaneko and Xu [11]. Moreover, the superconvergence of the iterated collocation method for Hammerstein equations with smooth as well as weakly singular kernels was studied by Kaneko et al [12].…”

“…For Hammerstein equations, generally, the iterated postprocessing method (see, for example [6,11,12,25]) is used to accelerate the approximation. This method is very effective but may be complicated in calculation.…”