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 Ih for delay Volterra integro-differential equations with proportional delay functions given by θ(t) = qt (0 < q < 1, t € [0, T]). In particular, we show that if Uh 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 - ε_N, where ε_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.