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Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ andthe integral operator $\mathcal{K}$ with kernel of the type of Green's functionare given. In the computation of approximate solutions of the given integralequation by Galerkin method, all the integrals are needed to be evaluated bysome numerical integration formula. This gives rise to the discrete version ofthe Galerkin method. For $r \geq 1$, a space of piecewise polynomials of degree$\leq r-1$ with respect to a uniform partition is chosen to be theapproximating space. For the appropriate choice of a numerical integrationformula, an asymptotic series expansion of the discrete iterated Galerkinsolution is obtained at the above partition points. Richardson extrapolation isused to improve the order of convergence. Using this method we can restore therate of convergence when the error is measured in the continuous case. Anumerical example is given to illustrate this theory.

Asymptotic error analysis for the discrete iterated Galerkin solution of Urysohn integral equations with Green's kernels