Residual-driven online generalized multiscale finite element methods

Handle URI:
http://hdl.handle.net/10754/577236
Title:
Residual-driven online generalized multiscale finite element methods
Authors:
Chung, Eric T.; Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Leung, Wing Tat
Abstract:
The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces.
KAUST Department:
Numerical Porous Media SRI Center (NumPor)
Citation:
Residual-driven online generalized multiscale finite element methods 2015 Journal of Computational Physics
Publisher:
Elsevier BV
Journal:
Journal of Computational Physics
Issue Date:
8-Sep-2015
DOI:
10.1016/j.jcp.2015.07.068
Type:
Article
ISSN:
00219991
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0021999115005744
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorChung, Eric T.en
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorLeung, Wing Taten
dc.date.accessioned2015-09-13T12:20:14Zen
dc.date.available2015-09-13T12:20:14Zen
dc.date.issued2015-09-08en
dc.identifier.citationResidual-driven online generalized multiscale finite element methods 2015 Journal of Computational Physicsen
dc.identifier.issn00219991en
dc.identifier.doi10.1016/j.jcp.2015.07.068en
dc.identifier.urihttp://hdl.handle.net/10754/577236en
dc.description.abstractThe construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0021999115005744en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, 8 September 2015. DOI: 10.1016/j.jcp.2015.07.068en
dc.subjectMultiscale finite element methoden
dc.subjectLocal model reductionen
dc.subjectAdaptivityen
dc.subjectOnline basis constructionen
dc.titleResidual-driven online generalized multiscale finite element methodsen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalJournal of Computational Physicsen
dc.eprint.versionPost-printen
dc.contributor.institutionDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong SARen
dc.contributor.institutionDepartment of Mathematics, Texas A&M University, College Station, TX, USAen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorEfendiev, Yalchin R.en
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