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Generalized multiscale finite element methods (GMsFEM)

Efendiev, Yalchin and Galvis, Juan and Hou, Thomas Y. (2013) Generalized multiscale finite element methods (GMsFEM). Journal of Computational Physics, 251 . pp. 116-135. ISSN 0021-9991. doi:10.1016/

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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.

Item Type:Article
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URLURL TypeDescription DOIArticle Paper
Efendiev, Yalchin0000-0001-9626-303X
Additional Information:© 2013 Elsevier Inc. Received 8 September 2012; Received in revised form 18 April 2013; Accepted 24 April 2013; Available online 22 May 2013. We would like to thank Ms. Guanglian Li for helping us with the computations and providing some computational results. Y. Efendiev’s work is partially supported by the DOE, US DoD Army ARO, and NSF (DMS 0934837 and DMS 0811180). J. Galvis would like to acknowledge partial support from DOE. This publication is based in part on work supported by Award No. KUSC1-016-04, made by King Abdullah University of Science and Technology (KAUST).
Funding AgencyGrant Number
Department of Energy (DOE)UNSPECIFIED
Army Research Office (ARO)UNSPECIFIED
NSFDMS 0934837
NSFDMS 0811180
King Abdullah University of Science and Technology (KAUST)KUS-C1-016-04
Subject Keywords:Multiscale; Input space; Proper orthogonal decomposition (POD); Local model reduction; Heterogeneous flow
Record Number:CaltechAUTHORS:20130830-131942975
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Official Citation:Yalchin Efendiev, Juan Galvis, Thomas Y. Hou, Generalized multiscale finite element methods (GMsFEM), Journal of Computational Physics, Volume 251, 15 October 2013, Pages 116-135, ISSN 0021-9991,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:41036
Deposited By: Tony Diaz
Deposited On:30 Aug 2013 20:55
Last Modified:10 Nov 2021 04:25

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