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One-parameter Galerkin Finite Element Methods for Neutral Reaction-diffusion Equations with Piecewise Continuous Arguments

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Abstract

This paper deals with Galerkin finite element (GFE) approximation to the initial-boundary value problems (IBVPs) of neutral reaction-diffusion equations with piecewise continuous arguments. For solving this kind of IBVPs, we first present a semi-discrete GFE scheme and give its error estimates in \(L^2\)- and \(H^1\)-norm. Then, we further construct a class of one-parameter fully discrete GFE methods with parameter \(\theta \) (\(0\! \le \! \theta \! \le \! 1\)) and analyze their unique solvability and \(L^2\)- and \(H^1\)-error. The result of error analysis shows that, under the suitable conditions and sense of \(L^2\)-norm (resp. \(H^1\)-norm), the one-parameter fully discrete GFE methods are convergent of order r (resp. \(r\! -\! 1\)) in space and order one (resp. two) in time when \(\theta \! \ne \! \frac{1}{2}\) (resp. \(\theta \! =\! \frac{1}{2}\)), where \(r\!-\!1~(r\! \ge \! 2)\) denotes the degree of the piecewise polynomial in finite element space. In the end, some numerical experiments are performed to verify the computational effectiveness and theoretical accuracy of the methods.

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All data and material generated or analysed during this study are included in this paper.

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Funding

The corresponding author Chengjian Zhang’s work is supported by NSFC (Grant No. 11971010).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074, Wuhan, China

    Hao Han & Chengjian Zhang

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, 430074, Wuhan, China

    Chengjian Zhang

Authors
  1. Hao Han
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  2. Chengjian Zhang
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Contributions

HH Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing-Original draft preparation, Visualization; CZ Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing-Review & Editing, Supervision, Project administration, Funding acquisition.

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Correspondence to Chengjian Zhang.

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This work is supported by NSFC (Grant No. 11971010)

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Han, H., Zhang, C. One-parameter Galerkin Finite Element Methods for Neutral Reaction-diffusion Equations with Piecewise Continuous Arguments. J Sci Comput 90, 91 (2022). https://doi.org/10.1007/s10915-022-01769-z

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  • DOI: https://doi.org/10.1007/s10915-022-01769-z

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