Collocation–quadrature methods and fast summation for Cauchy singular integral equations with fixed singularities

Abstract

Basing on recent results on the stability of collocation methods applied to Cauchy singular integral equations with additional fixed singularities we give necessary and sufficient conditions for the stability of collocation–quadrature methods for such equations. These methods have the advantage that the respective system of equations has a very simple structure and allows to apply fast summation methods which results in a fast algorithm with O(nlog⁡n) complexity. We present numerical results of the application of the proposed collocation–quadrature methods to the notched half plane problem of two-dimensional elasticity theory.