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

TL;DR

This paper analyzes the asymptotic error of the discrete iterated Galerkin method for solving Urysohn integral equations with Green's kernels, and demonstrates how Richardson extrapolation can improve convergence rates.

Contribution

It derives an asymptotic series expansion for the discrete Galerkin solution and applies Richardson extrapolation to enhance convergence in solving Urysohn integral equations.

Findings

Asymptotic series expansion of the discrete Galerkin solution is obtained.

Richardson extrapolation effectively improves convergence rates.

Numerical example confirms theoretical results.

Abstract

Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the 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 the approximating space. For the appropriate choice of a numerical integration formula, an asymptotic series expansion of the discrete iterated Galerkin solution is obtained at the above partition points. Richardson extrapolation is used to improve the order of convergence. Using this method we can restore the rate of convergence when the error is measured in the continuous case. A numerical example is given to illustrate this theory.