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An hp Version of the Discontinuous Galerkin Method for Volterra Integro-Differential Equations with Vanishing Delays

  • Lina Wang
  • Lijun YiEmail author
Article
  • 59 Downloads

Abstract

We present an hp version of the discontinuous Galerkin time stepping method for Volterra integro-differential equations with vanishing delays. We derive a priori error bounds in the \(L^2\)- and \(L^\infty \)-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Moreover, we prove that the hp version of the discontinuous 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. Numerical experiments are presented to illustrate the theoretical results.

Keywords

Volterra delay-integro-differential equations hp version Discontinuous Galerkin method Exponential rate of convergence 

Mathematics Subject Classification

65L60 65L05 65R20 65L70 

Notes

Acknowledgements

The authors would like to thank the two anonymous referees for many constructive and valuable suggestions, which considerably improved the presentation of the paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina

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