Generalized multiscale finite element methods: Oversampling strategies

Handle URI:
http://hdl.handle.net/10754/563235
Title:
Generalized multiscale finite element methods: Oversampling strategies
Authors:
Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Galvis, Juan; Li, Guanglian; Presho, Michael
Abstract:
In this paper, we propose oversampling strategies in the generalized multiscale finite element method (GMsFEM) framework. The GMsFEM, which has been recently introduced in Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) construction of the online space (the latter for parameter-dependent problems). In Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we develop oversampling techniques to be used in this context (see Hou and Wu (1997) where oversampling is introduced for multiscale finite element methods). It is known (see Hou and Wu (1997)) that the oversampling can improve the accuracy of multiscale methods. In particular, the oversampling technique uses larger regions (larger than the target coarse block) in constructing local basis functions. Our motivation stems from the analysis presented in this paper, which shows that when using oversampling techniques in the construction of the snapshot space and offline space, GMsFEM will converge independent of small scales and high contrast under certain assumptions. We consider the use of a multiple eigenvalue problems to improve the convergence and discuss their relation to single spectral problems that use oversampled regions. The oversampling procedures proposed in this paper differ from those in Hou and Wu (1997). In particular, the oversampling domains are partially used in constructing local spectral problems. We present numerical results and compare various oversampling techniques in order to complement the proposed technique and analysis.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Begell House
Journal:
International Journal for Multiscale Computational Engineering
Issue Date:
2014
DOI:
10.1615/IntJMultCompEng.2014007646
Type:
Article
ISSN:
15431649
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.authorLi, Guanglianen
dc.contributor.authorPresho, Michaelen
dc.date.accessioned2015-08-03T11:43:47Zen
dc.date.available2015-08-03T11:43:47Zen
dc.date.issued2014en
dc.identifier.issn15431649en
dc.identifier.doi10.1615/IntJMultCompEng.2014007646en
dc.identifier.urihttp://hdl.handle.net/10754/563235en
dc.description.abstractIn this paper, we propose oversampling strategies in the generalized multiscale finite element method (GMsFEM) framework. The GMsFEM, which has been recently introduced in Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) construction of the online space (the latter for parameter-dependent problems). In Efendiev et al. (2013b) [Generalized Multiscale Finite Element Methods, J. Comput. Phys., vol. 251, pp. 116-135, 2013], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we develop oversampling techniques to be used in this context (see Hou and Wu (1997) where oversampling is introduced for multiscale finite element methods). It is known (see Hou and Wu (1997)) that the oversampling can improve the accuracy of multiscale methods. In particular, the oversampling technique uses larger regions (larger than the target coarse block) in constructing local basis functions. Our motivation stems from the analysis presented in this paper, which shows that when using oversampling techniques in the construction of the snapshot space and offline space, GMsFEM will converge independent of small scales and high contrast under certain assumptions. We consider the use of a multiple eigenvalue problems to improve the convergence and discuss their relation to single spectral problems that use oversampled regions. The oversampling procedures proposed in this paper differ from those in Hou and Wu (1997). In particular, the oversampling domains are partially used in constructing local spectral problems. We present numerical results and compare various oversampling techniques in order to complement the proposed technique and analysis.en
dc.publisherBegell Houseen
dc.subjectGeneralized multiscale finite element methoden
dc.subjectHigh contrasten
dc.subjectOversamplingen
dc.titleGeneralized multiscale finite element methods: Oversampling strategiesen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalInternational Journal for Multiscale Computational Engineeringen
dc.contributor.institutionDepartment of Mathematics and Institute for Scientific Computation (ISC), Texas A and M UniversityCollege Station, TX, United Statesen
dc.contributor.institutionDepartamento de Matemáticas, Universidad Nacional de ColombiaBogotá D.C, Colombiaen
kaust.authorEfendiev, Yalchin R.en
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