Optimal Superconvergence Results for Delay Integro‐Differential Equations of Pantograph Type

Abstract

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 $\theta (t) = qt (0 < q < 1, t \in [0,T])$. In particular, we show that if $u_{h}$ is a continuous piecewise polynomial of degree $m \geq 2$, and if collocation is at the Gauss (-Legendre) points, then the (optimal) order of local superconvergence on $I_{h}$ is $p^{\ast } = m+2$. It turns out that the same order $p^{\ast }$ 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 $2m-\varepsilon _{N}$, where $\varepsilon _{N} \rightarrow 0$ 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.