Generalized multiscale finite element methods (GMsFEM)

Handle URI:
http://hdl.handle.net/10754/562987
Title:
Generalized multiscale finite element methods (GMsFEM)
Authors:
Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Galvis, Juan; Hou, Thomasyizhao
Abstract:
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. © 2013 Elsevier Inc.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Elsevier BV
Journal:
Journal of Computational Physics
Issue Date:
Oct-2013
DOI:
10.1016/j.jcp.2013.04.045
Type:
Article
ISSN:
00219991
Sponsors:
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. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorGalvis, Juanen
dc.contributor.authorHou, Thomasyizhaoen
dc.date.accessioned2015-08-03T11:18:13Zen
dc.date.available2015-08-03T11:18:13Zen
dc.date.issued2013-10en
dc.identifier.issn00219991en
dc.identifier.doi10.1016/j.jcp.2013.04.045en
dc.identifier.urihttp://hdl.handle.net/10754/562987en
dc.description.abstractIn 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. © 2013 Elsevier Inc.en
dc.description.sponsorshipWe 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. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherElsevier BVen
dc.subjectHeterogeneous flowen
dc.subjectInput spaceen
dc.subjectLocal model reductionen
dc.subjectMultiscaleen
dc.subjectProper orthogonal decomposition (POD)en
dc.titleGeneralized multiscale finite element methods (GMsFEM)en
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalJournal of Computational Physicsen
dc.contributor.institutionDepartment of Mathematics and ISC, Texas A and M University, College Station, TX 77843, United Statesen
dc.contributor.institutionISC, Texas A and M University, College Station, TX 77843, United Statesen
dc.contributor.institutionDepartamento de Matemáticas, Universidad Nacional de Colombia, Bogotá DC, Colombiaen
dc.contributor.institutionCaltech, Pasadena, CA 91125, United Statesen
kaust.authorEfendiev, Yalchin R.en
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