A Mixed Two-Grid Method Applied to a Fredholm Equation of the Second Kind

L. Grammont ()
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L. Grammont: Université de Saint-Étienne

Chapter 16 in Integral Methods in Science and Engineering, Volume 2, 2010, pp 173-181 from Springer

Abstract: Abstract The purpose of this chapter is to compute at a low cost an approximate solution of a Fredholm integral equation at a given accuracy. Vainikko proposed to compute the Nyström approximation of order n with quadrature two-grid iterations. We propose here to compute it with a two-grid method based on a projection method of a new type developed by Kulkarni. We will theoretically compare the absolute errors and the complexities of these two approximations. Let T be the integral operator $$Tu(t) = \int_0^1 k(t, x)u(x)dx,$$ where $$k \in C^{m^\prime}([0, 1]\times [0, 1]).$$ Consider the integral equation of the second kind 16.1 $$u - Tu = f,$$ where $$f \in C^m[0, 1]$$ . We will assume that this equation has a unique solution.

Date: 2010
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DOI: 10.1007/978-0-8176-4897-8_16

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