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Error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals

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Elliott, D and Johnston, PR (2007) Error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals. Journal of Computational and Applied Mathematics, 203 (1). pp. 103-124. ISSN 0377-0427

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Abstract

In the two-dimensional boundary element method, one often needs to evaluate numerically integrals of the form where j2 is a quadratic, g is a polynomial and f is a rational, logarithmic or algebraic function with a singularity at zero. The constants a and b are such that -1a1 and 0<b1 so that the singularities of f will be close to the interval of integration. In this case the direct application of Gauss–Legendre quadrature can give large truncation errors. By making the transformation x=a+bsinh(μu-η), where the constants μ and η are chosen so that the interval of integration is again [-1,1], it is found that the truncation errors arising, when the same Gauss–Legendre quadrature is applied to the transformed integral, are much reduced. The asymptotic error analysis for Gauss–Legendre quadrature, as given by Donaldson and Elliott [A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972) 573–602], is then used to explain this phenomenon and justify the transformation.

Item Type: Article
Keywords: Numerical integration; Boundary element method; Nearly singular integrals; Nonlinear coordinate transformation; Sinh function
Journal or Publication Title: Journal of Computational and Applied Mathematics
Publisher: Elsevier Science BV
Page Range: pp. 103-124
ISSN: 0377-0427
Identification Number - DOI: 10.1016/j.cam.2006.03.012
Date Deposited: 07 Apr 2008 14:06
Last Modified: 18 Nov 2014 03:33
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