A posteriori error estimation for a singularly perturbed Volterra integro-differential equation

A singularly perturbed Volterra integro-differential equation with an integrable singularity in the integral term is considered. The upwind difference method is used to discretize the singularly perturbed Volterra integro-differential equation. The stability properties and a posteriori error analysis for the discrete scheme are given. Since a priori information of the exact solution may be difficult to obtain, a posteriori adaptive algorithm based on a posteriori error analysis is established by equidistributing arc-length monitor function which attempts to automatically detect the singular behavior of the solution. Numerical experiments support this theoretical result.