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

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.