Iterated Collocation Methods and Their Discretizations for Volterra Integral Equations

Abstract

It is known that the numerical solution of Volterra integral equations of the second kind by polynomial spline collocation at the Gauss–Legendre points does not lead to local superconvergence at the knots of the approximating function. In the present paper we show that iterated collocation approximation restores optimal local superconvergence at the knots but does not yield global superconvergence on the entire interval of integration, in contrast to Fredholm integral equations with smooth kernels. We also analyze the discretized versions (obtained by suitable numerical quadrature) of the collocation and iterated collocation methods.