Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels

Abstract

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r\ge 1$, a space of piecewise polynomials of degree $\le r-1$ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x-\mathcal{K}(x)=f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.