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


Elliott, D and Johnston, PR, 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 (2007) [Refereed Article]

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DOI: doi:10.1016/


In the two-dimensional boundary element method, one often needs to evaluate numerically integrals of the form ∫- 1 1 g (x) j (x) f ((x - a)2 + b2) d x 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 - 1 ≤ a ≤ 1 and 0 < b ≪ 1 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 + b sinh (μ u - 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 Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Pure mathematics
Research Field:Ordinary differential equations, difference equations and dynamical systems
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the mathematical sciences
UTAS Author:Elliott, D (Professor David Elliott)
ID Code:48962
Year Published:2007
Web of Science® Times Cited:17
Deposited By:Mathematics
Deposited On:2007-08-01
Last Modified:2012-11-07
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