Extrapolation of a discrete collocation-type method of Hammerstein equations

In recent papers, Kumar and Sloan introduced a new collocation-type method for numerical solution of Hammerstein integral equations. Kumar studied a discretized version of this method and obtained superconvergence rate for the discrete approximation to the exact solution. In this paper, the asymptotic error expansion of a discrete collocation-type method for Hammerstein integral equations is obtained. We show that when piecewise polynomials of degree p − 1 are used and numerical quadrature is used to approximate the definite integrals occurring in this method, the approximation solution admits an error expansion in powers of the step-size h. For a special choice of collocation points and numerical quadrature rule, the leading terms in the error expansion for the collocation solution contain only even powers of the step-size h, beginning with a term h2p. Thus Richardsonʹs extrapolation can be performed on the solution, and this will increase the accuracy of numerical solution greatly. Some numerical results are given to illustrate this theory.