Exponentially Convergent Multiscale Finite Element Method

doi:10.1007/s42967-023-00260-2

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (2): 862-878.doi: 10.1007/s42967-023-00260-2

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Exponentially Convergent Multiscale Finite Element Method

Yifan Chen, Thomas Y. Hou, Yixuan Wang

Applied and Computational Mathematics, Caltech, Pasadena 91106, USA

收稿日期:2022-12-01 修回日期:2022-12-01 接受日期:2023-02-05 出版日期:2023-05-03 发布日期:2023-05-03
通讯作者: Yifan Chen,E-mail:yifanc@caltech.edu;Thomas Y. Hou,E-mail:hou@cms.caltech.edu;Yixuan Wang,E-mail:roywang@caltech.edu E-mail:yifanc@caltech.edu;hou@cms.caltech.edu;roywang@caltech.edu
基金资助:
This research is in part supported by the NSF Grants DMS-1912654 and DMS 2205590.

Exponentially Convergent Multiscale Finite Element Method

Yifan Chen, Thomas Y. Hou, Yixuan Wang

Applied and Computational Mathematics, Caltech, Pasadena 91106, USA

Received:2022-12-01 Revised:2022-12-01 Accepted:2023-02-05 Online:2023-05-03 Published:2023-05-03
Contact: Yifan Chen,E-mail:yifanc@caltech.edu;Thomas Y. Hou,E-mail:hou@cms.caltech.edu;Yixuan Wang,E-mail:roywang@caltech.edu E-mail:yifanc@caltech.edu;hou@cms.caltech.edu;roywang@caltech.edu
Supported by:
This research is in part supported by the NSF Grants DMS-1912654 and DMS 2205590.

摘要/Abstract

摘要： We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

关键词: Multiscale method, Exponential convergence, Helmholtz's equation, Domain decomposition, Nonlinear model reduction

Abstract: We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

Key words: Multiscale method, Exponential convergence, Helmholtz's equation, Domain decomposition, Nonlinear model reduction

Yifan Chen, Thomas Y. Hou, Yixuan Wang. Exponentially Convergent Multiscale Finite Element Method[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 862-878.

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Exponentially Convergent Multiscale Finite Element Method

Exponentially Convergent Multiscale Finite Element Method

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