Huang, Jin; Lü, Tao; Zhu, Rui High accuracy combination algorithms for solving singular integral equations with Hilbert kernel by quadrature methods. (Chinese. English summary) Zbl 1199.65419 Acta Math. Sci., Ser. A, Chin. Ed. 29, No. 1, 103-113 (2009). Summary: 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 \(O(e^{-n\delta})\) if the coefficients of equations belong to \(B_\delta\). Besides, using the combination algorithms can not only obtain a higher order of accuracy, but also a posterior error estimate is deduced. These excellent numerical results display the significance of the algorithms proposed in the paper. MSC: 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 65Y05 Parallel numerical computation Keywords:Hilbert singular integral equation; a posterior error estimate; quadrature method; parallel computation; Hilbert kernel; algorithms; numerical results PDFBibTeX XMLCite \textit{J. Huang} et al., Acta Math. Sci., Ser. A, Chin. Ed. 29, No. 1, 103--113 (2009; Zbl 1199.65419)