Journal of Scientific Computing

, Volume 74, Issue 2, pp 1091–1114 | Cite as

An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations

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Abstract

We investigate an h-p version of the continuous Petrov–Galerkin time stepping method for nonlinear delay differential equations with vanishing delays. We derive a priori error estimates in the \(L^{2}\)-, \(H^{1}\)- and \(L^\infty \)-norm that are completely explicit with respect to the local time steps, the local polynomial degrees, and the local regularity of the exact solution. Moreover, we show that the h-p version continuous Petrov–Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. The theoretical results are illustrated by some numerical experiments.

Keywords

Nonlinear delay differential equations h-p version Continuous Petrov–Galerkin method Error analysis 

Mathematics Subject Classification

65L60 65L05 65L70 

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina

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