Modified projection and the iterated modified projection methods for nonlinear integral equations

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 $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution and $2 r + 2$ for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order $ 3 r + 3$ and one step of iteration is shown to improve the order of convergence to $ 4 r + 4$. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.