# Richardson extrapolation for the discrete iterated modified projection solution

## TL;DR

This paper develops an asymptotic expansion for the discrete iterated modified projection solution of Urysohn integral equations and employs Richardson extrapolation to enhance convergence order, ensuring numerical quadrature preserves the continuous case's convergence rate.

## Contribution

It introduces an asymptotic expansion for the discrete iterated modified projection solution and applies Richardson extrapolation to improve convergence order in numerical solutions.

## Key findings

- Richardson extrapolation effectively increases convergence order.
- A suitable numerical quadrature preserves the continuous case's convergence rate.
- Asymptotic expansion provides insights into the discrete solution's behavior.

## 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.