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Article: On the convergence rates of GMSFEMs for heterogeneous elliptic problems without oversampling techniques

TitleOn the convergence rates of GMSFEMs for heterogeneous elliptic problems without oversampling techniques
Authors
KeywordsGMsFEM
Heterogeneous coefficient
High-contrast
Multiscale methods
Spectral basis
Harmonic extension basis functions
Issue Date2019
Citation
Multiscale Modeling and Simulation, 2019, v. 17, n. 2, p. 593-619 How to Cite?
Abstract\bigcirc c 2019 Society for Industrial and Applied Mathematics This work is concerned with the rigorous analysis of the generalized multiscale finite element methods (GMsFEMs) for elliptic problems with high-contrast heterogeneous coefficients. GMsFEMs are popular numerical methods for solving flow problems with heterogeneous high-contrast coefficients, and they have demonstrated extremely promising numerical results for a wide range of applications. However, the mathematical justification of the efficiency of the method is still largely missing. In this work, we analyze two types of multiscale basis functions, i.e., local spectral basis functions and basis functions of local harmonic extension type, within the GMsFEM framework. These constructions have found many applications in the past few years. We establish their optimal convergence in the energy norm or H1 seminorm under a very mild assumption that the source term belongs to some weighted L2 space, and without the help of any oversampling technique. Furthermore, we analyze the model order reduction of the local harmonic extension basis and prove its convergence in the energy norm. These theoretical findings offer insight into the mechanism behind the efficiency of the GMsFEMs.
Persistent Identifierhttp://hdl.handle.net/10722/287061
ISSN
2019 Impact Factor: 1.855
2015 SCImago Journal Rankings: 1.062
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorLi, Guanglian-
dc.date.accessioned2020-09-07T11:46:24Z-
dc.date.available2020-09-07T11:46:24Z-
dc.date.issued2019-
dc.identifier.citationMultiscale Modeling and Simulation, 2019, v. 17, n. 2, p. 593-619-
dc.identifier.issn1540-3459-
dc.identifier.urihttp://hdl.handle.net/10722/287061-
dc.description.abstract\bigcirc c 2019 Society for Industrial and Applied Mathematics This work is concerned with the rigorous analysis of the generalized multiscale finite element methods (GMsFEMs) for elliptic problems with high-contrast heterogeneous coefficients. GMsFEMs are popular numerical methods for solving flow problems with heterogeneous high-contrast coefficients, and they have demonstrated extremely promising numerical results for a wide range of applications. However, the mathematical justification of the efficiency of the method is still largely missing. In this work, we analyze two types of multiscale basis functions, i.e., local spectral basis functions and basis functions of local harmonic extension type, within the GMsFEM framework. These constructions have found many applications in the past few years. We establish their optimal convergence in the energy norm or H1 seminorm under a very mild assumption that the source term belongs to some weighted L2 space, and without the help of any oversampling technique. Furthermore, we analyze the model order reduction of the local harmonic extension basis and prove its convergence in the energy norm. These theoretical findings offer insight into the mechanism behind the efficiency of the GMsFEMs.-
dc.languageeng-
dc.relation.ispartofMultiscale Modeling and Simulation-
dc.subjectGMsFEM-
dc.subjectHeterogeneous coefficient-
dc.subjectHigh-contrast-
dc.subjectMultiscale methods-
dc.subjectSpectral basis-
dc.subjectHarmonic extension basis functions-
dc.titleOn the convergence rates of GMSFEMs for heterogeneous elliptic problems without oversampling techniques-
dc.typeArticle-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1137/18M1172715-
dc.identifier.scopuseid_2-s2.0-85068438092-
dc.identifier.volume17-
dc.identifier.issue2-
dc.identifier.spage593-
dc.identifier.epage619-
dc.identifier.eissn1540-3467-
dc.identifier.isiWOS:000473063800001-
dc.identifier.issnl1540-3459-

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