Richardson extrapolation for the discrete iterated modified projection solution

Abstract

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.