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The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle
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The superconvergence of composite Newton–Cotes rules for Hadamard finite-part integral on a circle
Xiaoping Zhang1, 3  , Jiming Wu2 and Dehao Yu3
| (1) |
School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China |
| (2) |
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, 100088 Beijing, People’s Republic of China |
| (3) |
LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, CAS, 100190 Beijing, People’s Republic of China |
Received: 2 January 2009 Accepted: 3 June 2009 Published online: 17 June 2009
Communicated by W. Hackbusch.
Abstract We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral on a circle with
the hypersingular kernel  and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known
point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite)
Newton–Cotes rules occurs at the zeros of a special function  and prove the existence of the superconvergence points. The relation between  and  defined in Wu and Sun (Numer Math 109:143–165, 2008) is established, and the efficient calculation of Cotes coefficients
is also discussed. Several numerical examples are provided to validate the theoretical analysis.
Keywords Hadamard finite-part integral - Composite Newton–Cotes rule - Superconvergence - Clausen function
Mathematics Subject Classification (2000) 65D30 - 65D32
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