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The Uniform Convergence of a DG Method for a Singularly Perturbed Volterra Integro-Differential Equation

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Abstract

The purpose of this work is to implement a discontinuous Galerkin (DG) method with a one-sided flux for a singularly perturbed Volterra integro-differential equation (VIDE) with a smooth kernel. First, the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided. Then the existence and uniqueness of the DG solution are proven. Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established. Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants, the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter ϵ when the space of polynomials with degree p is used. A numerical experiment validates the theoretical results. Furthermore, an ultra-convergence order 2p + 1 at the nodes for the one-sided flux, uniform with respect to the singular perturbation parameter ϵ, is observed numerically.

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Authors and Affiliations

  1. School of Mathematics, Hunan Institute of Science and Technology, Yueyang, 414006, China

    Xia Tao

  2. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Xia Tao & Ziqing Xie

Authors
  1. Xia Tao
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  2. Ziqing Xie
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Correspondence to Ziqing Xie.

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Conflict of Interest

Ziqing XIE is an editorial board member for Acta Mathematica Scientia and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.

Tao’s research was supported by the National Natural Science Foundation of China (12001189). Xie’s research was supported by the National Natural Science Foundation of China (11171104, 12171148).

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Tao, X., Xie, Z. The Uniform Convergence of a DG Method for a Singularly Perturbed Volterra Integro-Differential Equation. Acta Math Sci 43, 2159–2178 (2023). https://doi.org/10.1007/s10473-023-0514-8

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  • DOI: https://doi.org/10.1007/s10473-023-0514-8

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