  We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type
of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree
$\leq 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.
