An h–p Version of the Discontinuous Galerkin Method for Volterra Integro-Differential Equations with Vanishing Delays
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We present an h–p 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 h–p 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.
KeywordsVolterra delay-integro-differential equations h–p version Discontinuous Galerkin method Exponential rate of convergence
Mathematics Subject Classification65L60 65L05 65R20 65L70
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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