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An \(h\)–\(p\) Version of the Continuous Petrov–Galerkin Finite Element Method for Nonlinear Volterra Integro-Differential Equations

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Abstract

We present an \(h\)–\(p\) version of the continuous Petrov–Galerkin finite element method for nonlinear Volterra integro-differential equations. We derive a priori error bounds in the \(L^2\)- and \(H^1\)-norm that are explicit in the time steps, the approximation orders, and the regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.

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Acknowledgments

The author would like to thank the referee for many constructive and valuable suggestions, which considerably improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lijun Yi

  2. Division of Computational Science, E-institute of Shanghai Universities, Shanghai, 200234, China

    Lijun Yi

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  1. Lijun Yi
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Correspondence to Lijun Yi.

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Dedicated to Professor Benqi Guo on the Occasion of his 70th Birthday.

The work of this author is supported in part by the NSF of China (Nos. 11301343 and 11226330), the Research Fund for the Doctoral Program of Higher Education of China (No. 20113127120002), the Research Fund for Young Teachers Program in Shanghai (No. shsf018), and the Fund for E-institute of Shanghai Universities (No. E03004).

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Yi, L. An \(h\)–\(p\) Version of the Continuous Petrov–Galerkin Finite Element Method for Nonlinear Volterra Integro-Differential Equations. J Sci Comput 65, 715–734 (2015). https://doi.org/10.1007/s10915-015-9983-9

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  • DOI: https://doi.org/10.1007/s10915-015-9983-9

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