Multilevel correction for collocation solutions of Volterra integral equations with proportional delays

In this paper we propose a convergence acceleration method for collocation solutions of the linear second-kind Volterra integral equations with proportional delay qt (0 < q < 1). This convergence acceleration method called multilevel correction method is based on a kind of hybrid mesh, which can be viewed as a combination between the geometric meshes and the uniform meshes. It will be shown that, when the collocation solutions are continuous piecewise polynomials whose degrees are less than or equal to m (m 6 2), the global accuracy of k level corrected approximation is O(N−(2m(k+1)−ε)), where N is the number of the nodes, and ε is an arbitrary small positive number.