Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels
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 r1 $ 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.08879
 Bibcode:
 2020arXiv201208879R
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Functional Analysis;
 45G10;
 65B05;
 65J15;
 65R20
 EPrint:
 International Journal of computer Mathematics, 2021