The superconvergence of composite trapezoidal rule for Hadamard finite-part integral on a circle and its application

Abstract

The quadrature method for Hadamard finite-part integral on a circle is discussed and the emphasis is placed on the pointwise superconvergence phenomenon of the composite trapezoidal rule, i.e. when the singular point coincides with some a priori known points, the accuracy can be better than what is globally possible. The existence and uniqueness of the superconvergence points are proved and the correspondent superconvergence estimate is obtained. An indirect method is introduced and then applied to solve the integral equation of the second kind containing finite-part kernels, including that arising in the scattering theory. Some numerical results are also presented to confirm the theoretical results and to show the efficiency of the algorithms.

Keywords:

• finite-part integral
• composite trapezoidal rule
• superconvergence
• finite-part integral equation of the second kind
• scattering problem

2000 AMS Subject Classification :

• 65D30
• 65D32
• 45E99

Acknowledgements

The work of X.P. Zhang and D.H. Yu was supported in part by the National Basic Research Program of China (No. 2005CB321701), the National Natural Science Foundation of China (No. 10531080) and the Natural Science Foundation of Beijing (No. 1072009). The work of J.M. Wu was supported by the National Natural Science Foundation of China (No. 10671025).