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Unconditionally Energy-Stable Finite Element Scheme for the Chemotaxis-Fluid System

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Abstract

In this paper, we first deduce an improved chemotaxis-fluid system by introducing a chemotactic stress force, which can be used to describe the chemotactic movement of bacteria in a viscous fluid. Compared with the classical chemotaxis-Navier–Stokes system, the newly modified system obeys the law of energy dissipation. To solve such a chemotaxis-fluid system, we develop a linear, decoupled fully-discrete finite element scheme by combining the scalar auxiliary variable (SAV) approach, implicit-explicit (IMEX) scheme and pressure-projection method. The unconditional energy stability of the developed scheme is proved rigorously, and we further prove the optimal error estimates for the fully discrete scheme, especially for the pressure bound. Finally, some numerical examples are presented to verify the accuracy, energy stability and performance of the proposed numerical scheme.

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Acknowledgements

The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. The work of Guang-an Zou is supported by China Postdoctoral Science Foundation (No.2019M662476), and the Key Scientific Research Projects of Colleges and Universities in Henan Province, China (23A110006). Jian Li is supported by NSF of China (No.11771259), Shaanxi Provincial Joint Laboratory of Artificial Intelligence (No.2022JC-SYS-05), Innovative team project of Shaanxi Provincial Department of Education(No.21JP013) and 2022 Shaanxi Provincial Social Science Fund Annual Project (No.2022D332).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Yangyang Tang & Guang-an Zou

  2. Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, People’s Republic of China

    Guang-an Zou

  3. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi’an, 710021, People’s Republic of China

    Jian Li

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  1. Yangyang Tang
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Tang, Y., Zou, Ga. & Li, J. Unconditionally Energy-Stable Finite Element Scheme for the Chemotaxis-Fluid System. J Sci Comput 95, 1 (2023). https://doi.org/10.1007/s10915-023-02118-4

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