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Exponentially Convergent Multiscale Finite Element Method

Chen, Yifan and Hou, Thomas Y. and Wang, Yixuan (2022) Exponentially Convergent Multiscale Finite Element Method. . (Unpublished)

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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. 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 MsFEM in the literature, 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 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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Chen, Yifan0000-0001-5494-4435
Hou, Thomas Y.0000-0001-6287-1133
Wang, Yixuan0000-0001-7305-5422
Additional Information:This research is in part supported by NSF Grants DMS-1912654 and DMS 2205590. We would also like to acknowledge the generous support from Mr. K. C. Choi through the Choi Family Gift Fund. The authors have no other relevant financial or non-financial interests to disclose.
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Choi Family Gift FundUNSPECIFIED
Record Number:CaltechAUTHORS:20230227-194420642
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:119521
Deposited By: George Porter
Deposited On:28 Feb 2023 03:35
Last Modified:28 Feb 2023 03:35

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