  Approximate solutions of Urysohn integral equations using projection methods
involve integrals which need to be evaluated using a numerical quadrature
formula. It gives rise to the discrete versions of the projection methods. For
$r \geq 1,$ a space of piece-wise polynomials of degree $\leq r - 1$ with
respect to an uniform partition is chosen to be the approximating space and the
projection is chosen to be the interpolatory projection at $r$ Gauss points.
Asymptotic expansion for the iterated modified projection solution is available
in literature. In this paper, we obtain an asymptotic expansion for the
discrete iterated modified projection solution and use Richardson extrapolation
to improve the order of convergence. Our results indicate a choice of a
numerical quadrature which preserves the order of convergence in the continuous
case.
