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

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Author(s)
Elliott, David
Johnston, Peter R
Griffith University Author(s)
Year published
2007
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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) dx where j 2 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( u- ), where the constants and are chosen so that the interval of integration is again ...
View more >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) dx where j 2 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( u- ), where the constants and 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.
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View more >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) dx where j 2 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( u- ), where the constants and 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.
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Journal Title
Journal of Computational and Applied Mathematics
Volume
203
Issue
1
Publisher URI
Copyright Statement
© 2007 Elsevier. This is the author-manuscript version of this paper. Reproduced in accordance with the copyright policy of the publisher. Please refer to the journal's website for access to the definitive, published version.
Subject
Applied mathematics
Numerical and computational mathematics