## Abstract

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.

Original language | English |
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Pages (from-to) | 986-1004 |

Number of pages | 19 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 45 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 |

## Scopus Subject Areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

- Collocation solutions
- Optimal order of superconvergence
- Pantograph equation
- Proportional delays
- Vanishing delays
- Volterra integro-differential equation