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Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems

Author

Listed:
  • Shan Jiang

    (School of Science, Nantong University, Nantong 226019, China)

  • Yue Cheng

    (School of Science, Nantong University, Nantong 226019, China)

  • Yao Cheng

    (School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China)

  • Yunqing Huang

    (School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
    Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan 411105, China
    Key Laboratory of Intelligent Computing Information Processing of Ministry of Education, Xiangtan 411105, China)

Abstract

We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ -implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method.

Suggested Citation

  • Shan Jiang & Yue Cheng & Yao Cheng & Yunqing Huang, 2023. "Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems," Mathematics, MDPI, vol. 11(24), pages 1-14, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4965-:d:1300678
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    References listed on IDEAS

    as
    1. Denis Spiridonov & Jian Huang & Maria Vasilyeva & Yunqing Huang & Eric T. Chung, 2019. "Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model," Mathematics, MDPI, vol. 7(12), pages 1-13, December.
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