Optimal superconvergence results for delay integro-differential equations of pantograph type

We analyze the optimal ( global and local) orders of superconvergence of collocation solutions u(h) on uniform meshes I-h for delay Volterra integro-differential equations with proportional delay functions given by theta(t) = qt (0 < q < 1, t epsilon [ 0, T]). In particular, we show that if u(h) is a continuous piecewise polynomial of degree m >= 2, and if collocation is at the Gauss (-Legendre) points, then the ( optimal) order of local superconvergence on I-h is p* = m + 2. It turns out that the same order p* holds for nonlinear ( strictly increasing) delay functions vanishing at t = 0. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order 2m - epsilon(N), where epsilon N -> 0 as the number N of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis.