Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution, $2 r + 2$ for the iterated Galerkin solution, $ 3 r + 3$ for the modified projection solution and $ 4 r + 4$ for the iterated modified projection solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, the iterated Galerkin/iterated collocation solution and the iterated modified projection solution are shown to have asymptotic series expansions. The Richardson extrapolation can then be used to improve the order of convergence to $ 2 r + 4$ in the case of the iterated Galerkin/iterated collocation method and to $ 4 r + 6$ in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence.
"Asymptotic error analysis of projection and modified projection methods for nonlinear integral equations." J. Integral Equations Applications 27 (1) 67 - 101, SPRING 2015. https://doi.org/10.1216/JIE-2015-27-1-67