Abstract: This paper presents high accuracy combination algorithms for solving singular integral equations with Hilbert kernel by quadrature methods. A given fine grid set is divided into some subsets with different grid points. After these discrete equations dependent on the subsets are solved in parallel, the global fine grid approximations can be computed by the combination algorithms. It shows that the accuracy of quadrature methods is very high with O(e^-nδ) if the coefficients of equations belong to B_δ, Besides, using the combination algorithms can not only obtain a higher order of the accuracy, but also a posterior error estimate is deduced. These excellent numerical results display the significance of these algorithms made in the paper.

Key words: Hilbert singular integral equation, Combination algorithm, A posterior error estimate, Quadrature method.