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

Shan Jiang (), Yue Cheng, Yao Cheng and Yunqing Huang
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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

Mathematics, 2023, vol. 11, issue 24, 1-14

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.

Keywords: generalized multiscale finite element; balanced truncation; eigenvalue computation; fully discretization; uniform stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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