No abstract

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 ≤ r, previous authors have established an order r + 1 convergence for the Galerkin solution, 2r + 2 for the iterated Galerkin solution, 3r + 3 for the modified projection solution and 4r + 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 2r + 4 in the case of the iterated Galerkin/iterated collocation method and to 4r + 6 in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence. 2010 AMS Mathematics subject classification. Primary 45L10, 65J15, 65R20. Keywords and phrases. Urysohn integral operator, Galerkin method, collocation at Gauss points, extrapolation.

Consider a nonlinear operator equation x − K(x) = f , where K is a Urysohn integral operator with a kernel of the type of Green's function and defined on L ∞ [0, 1]. For r ≥ 0, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree ≤ r with respect to a quasi-uniform partition of [0, 1] and consider an interpolatory projection at r + 1 Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are r + 1 and r + 2 + min{r, 1}, respectively. We show that the order of convergence in the iterated modified projection method is 4 if r = 0 and is 2r + 3 if r ≥ 1. This improvement in the order of convergence is achieved while retaining the size of the system of equations that needs to be solved, the same as in the case of the collocation method. Numerical results are given for specific examples. 2010 AMS Mathematics subject classification. Primary 45L10, 65J15, 65R20. Keywords and phrases. Urysohn integral operator, collocation method, Gauss points.The first author would like to thank the Indo-French Centre for Applied Mathematics (IFCAM), Bangalore, India, for the partial support.