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A posteriori error estimation for a singularly perturbed Volterra integro-differential equation

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Abstract

A singularly perturbed Volterra integro-differential equation with an integrable singularity in the integral term is considered. The upwind difference method is used to discretize the singularly perturbed Volterra integro-differential equation. The stability properties and a posteriori error analysis for the discrete scheme are given. Since a priori information of the exact solution may be difficult to obtain, a posteriori adaptive algorithm based on a posteriori error analysis is established by equidistributing arc-length monitor function which attempts to automatically detect the singular behavior of the solution. Numerical experiments support this theoretical result.

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Acknowledgements

We would like to thank the anonymous reviewers for some suggestions for the improvement of this paper.

Funding

The work was supported by the Project of Philosophy and Social Science Research in Zhejiang Province (Grant No. 19NDJC039Z), Humanities and Social Sciences Planning Fund of Ministry of Education of China (Grant No. 18YJAZH002), Zhejiang Provincial Public Welfare Project of China (Grant No. LGF19A010001), Major Humanities and Social Sciences Projects in Colleges and Universities of Zhejiang Province (Grant No. 2018GH020), Ningbo Municipal Soft Science Foundation (Grant No. 2018A10041), Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010001), National Natural Science Foundation of China (Grant No. 11761015), and Natural Science Foundation of Guangxi (Grant No. 2017GXNSFBA198183).

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Authors and Affiliations

  1. Institute of Mathematics, Zhejiang Wanli University, Ningbo, 315100, Zhejiang, People’s Republic of China

    Jian Huang, Zhongdi Cen & Aimin Xu

  2. School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, 530023, Guangxi, People’s Republic of China

    Li-Bin Liu

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  1. Jian Huang
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  2. Zhongdi Cen
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  3. Aimin Xu
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  4. Li-Bin Liu
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Correspondence to Zhongdi Cen.

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Huang, J., Cen, Z., Xu, A. et al. A posteriori error estimation for a singularly perturbed Volterra integro-differential equation. Numer Algor 83, 549–563 (2020). https://doi.org/10.1007/s11075-019-00693-y

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