Asymptotic estimation of Gaussian quadrature error for a nonsingular integral in potential theory

Abstract:

This paper considers the $n$-point Gauss-Jacobi approximation of nonsingular integrals of the form $\int _{-1}^1 \mu (t) \phi (t) \log (z-t) \mathrm {d}t$, with Jacobi weight $\mu$ and polynomial $\phi$, and derives an estimate for the quadrature error that is asymptotic as $n \to \infty$. The approach follows that previously described by Donaldson and Elliott. A numerical example illustrating the accuracy of the asymptotic estimate is presented. The extension of the theory to similar integrals defined on more general analytic arcs is outlined.