Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels

TL;DR

This paper applies Richardson extrapolation to the iterated Galerkin method for solving Urysohn integral equations with Green's kernels, enhancing convergence order and providing theoretical and numerical validation.

Contribution

It introduces a novel application of Richardson extrapolation to improve the convergence of the iterated Galerkin method for Urysohn integral equations with Green's kernels.

Findings

Convergence order is improved using Richardson extrapolation.

Asymptotic expansion at partition points is derived.

Numerical example confirms theoretical improvements.

Abstract

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.