Open Access
SPRING 2015 Asymptotic error analysis of projection and modified projection methods for nonlinear integral equations
Rekha P. Kulkarni, T.J. Nidhin
J. Integral Equations Applications 27(1): 67-101 (SPRING 2015). DOI: 10.1216/JIE-2015-27-1-67


Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r$, previous authors have established an order $r + 1$ convergence for the Galerkin solution, $2 r + 2$ for the iterated Galerkin solution, $ 3 r + 3$ for the modified projection solution and $ 4 r + 4$ for the iterated modified projection solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, the iterated Galerkin/iterated collocation solution and the iterated modified projection solution are shown to have asymptotic series expansions. The Richardson extrapolation can then be used to improve the order of convergence to $ 2 r + 4$ in the case of the iterated Galerkin/iterated collocation method and to $ 4 r + 6$ in the case of the iterated modified projection method. Numerical results are given to illustrate this improvement in the orders of convergence.


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Rekha P. Kulkarni. T.J. Nidhin. "Asymptotic error analysis of projection and modified projection methods for nonlinear integral equations." J. Integral Equations Applications 27 (1) 67 - 101, SPRING 2015.


Published: SPRING 2015
First available in Project Euclid: 24 February 2015

zbMATH: 1328.45010
MathSciNet: MR3316979
Digital Object Identifier: 10.1216/JIE-2015-27-1-67

Primary: 45L10 , 65J15 , 65R20

Keywords: collocation at Gauss points , extrapolation , Galerkin method , Urysohn integral operator

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

Vol.27 • No. 1 • SPRING 2015
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