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Nonlinear Galerkin finite element methods for fourth-order Bi-flux diffusion model with nonlinear reaction term

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Abstract

A fourth-order diffusion model is presented with a nonlinear reaction term to simulate some special chemical and biological phenomenon. To obtain the solutions to those problems, the nonlinear Galerkin finite element method under the framework of the Hermite polynomial function for the spatial domain is utilized. The Euler backward difference method is used to solve the equation in the temporal domain. Subject to the Dirichlet and Navier boundary conditions, the numerical experiments for Bi-flux Fisher–Kolmogorov model present excellent convergence, accuracy and acceleration behavior. Also, the numerical solutions to the Bi-flux Gray–Scott model, subject to no flux boundary conditions, show excellent convergence, accuracy and symmetry.

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Acknowledgements

Jiang Zhu’s work was partially supported by the National Council for Scientific and Technological Development of Brazil (CNPq). Xijun Yu’s work was supported partially by the National Natural Science Foundation of China (Grant No. 11571002, 11772067, 11702028) and CAEP Foundation of China (Grant No. CX2019032). Luiz Bevilacqua’s work was supported partially by CNPq/TWAS Grant, the COPPE/CAPES Grant 001, and the USP/IEA visiting research program. Maosheng Jiang’s work was supported partially by CNPq/TWAS Grant.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

    Maosheng Jiang

  2. Alberto Luiz Coimbra Institute Graduate School of Engineering (COPPE), Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Luiz Bevilacqua

  3. Laboratório Nacional de Computação Científica, MCTIC, Avenida Getúlio Vargas 333, Petrópolis, RJ, 25651-075, Brazil

    Jiang Zhu

  4. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Xijun Yu

  5. Institute for Advanced Studies, University of São Paulo, São Paulo, Brazil

    Luiz Bevilacqua

Authors
  1. Maosheng Jiang
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  2. Luiz Bevilacqua
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  3. Jiang Zhu
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  4. Xijun Yu
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Corresponding author

Correspondence to Luiz Bevilacqua.

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Communicated by Abimael Loula.

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Jiang, M., Bevilacqua, L., Zhu, J. et al. Nonlinear Galerkin finite element methods for fourth-order Bi-flux diffusion model with nonlinear reaction term. Comp. Appl. Math. 39, 143 (2020). https://doi.org/10.1007/s40314-020-01168-w

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  • DOI: https://doi.org/10.1007/s40314-020-01168-w

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