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dc.contributor.authorEfendiev, Yalchin R.
dc.contributor.authorGalvis, Juan
dc.contributor.authorHou, Thomasyizhao
dc.date.accessioned2015-08-03T11:18:13Z
dc.date.available2015-08-03T11:18:13Z
dc.date.issued2013-10
dc.identifier.issn00219991
dc.identifier.doi10.1016/j.jcp.2013.04.045
dc.identifier.urihttp://hdl.handle.net/10754/562987
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.
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).
dc.publisherElsevier BV
dc.subjectHeterogeneous flow
dc.subjectInput space
dc.subjectLocal model reduction
dc.subjectMultiscale
dc.subjectProper orthogonal decomposition (POD)
dc.titleGeneralized multiscale finite element methods (GMsFEM)
dc.typeArticle
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.identifier.journalJournal of Computational Physics
dc.contributor.institutionDepartment of Mathematics and ISC, Texas A and M University, College Station, TX 77843, United States
dc.contributor.institutionISC, Texas A and M University, College Station, TX 77843, United States
dc.contributor.institutionDepartamento de Matemáticas, Universidad Nacional de Colombia, Bogotá DC, Colombia
dc.contributor.institutionCaltech, Pasadena, CA 91125, United States
dc.identifier.arxividarXiv:1301.2866
kaust.personEfendiev, Yalchin R.
kaust.grant.numberKUS-C1-016-04


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