Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media

Handle URI:
http://hdl.handle.net/10754/555672
Title:
Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media
Authors:
Chung, Eric T.; Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Leung, Wing Tat
Abstract:
Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to their complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the generalized multiscale finite element method (GMsFEM) (see [Y. Efendiev, J. Galvis, and T. Hou, J. Comput. Phys., 251 (2012), pp. 116--135]). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block, where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral problems are different from each other and their formulations are based on the analysis. To the best of knowledge, this is the first time that multiple snapshot spaces and multiple spectral problems are used and necessary for efficient computations. Using the dominant modes from local spectral problems, multiscale basis functions are constructed to represent the solution space locally within each coarse block. These multiscale basis functions are coupled via the symmetric interior penalty discontinuous Galerkin method which provides a block diagonal mass matrix and, consequently, results in fast computations in an explicit time discretization. Our methods' stability and spectral convergence are rigorously analyzed. Numerical examples are presented to show our methods' performance. We also test oversampling strategies. In particular, we discuss how the modes from different snapshot spaces can affect the proposed methods' accuracy.
KAUST Department:
Numerical Porous Media SRI Center (NumPor)
Citation:
Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media 2014, 12 (4):1691 Multiscale Modeling & Simulation
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
Multiscale Modeling & Simulation
Issue Date:
13-Nov-2014
DOI:
10.1137/130926675
Type:
Article
ISSN:
1540-3459; 1540-3467
Additional Links:
http://epubs.siam.org/doi/abs/10.1137/130926675
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.accessioned2015-05-25T08:34:50Zen
dc.date.available2015-05-25T08:34:50Zen
dc.date.issued2014-11-13en
dc.identifier.citationGeneralized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media 2014, 12 (4):1691 Multiscale Modeling & Simulationen
dc.identifier.issn1540-3459en
dc.identifier.issn1540-3467en
dc.identifier.doi10.1137/130926675en
dc.identifier.urihttp://hdl.handle.net/10754/555672en
dc.description.abstractNumerical modeling of wave propagation in heterogeneous media is important in many applications. Due to their complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the generalized multiscale finite element method (GMsFEM) (see [Y. Efendiev, J. Galvis, and T. Hou, J. Comput. Phys., 251 (2012), pp. 116--135]). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block, where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral problems are different from each other and their formulations are based on the analysis. To the best of knowledge, this is the first time that multiple snapshot spaces and multiple spectral problems are used and necessary for efficient computations. Using the dominant modes from local spectral problems, multiscale basis functions are constructed to represent the solution space locally within each coarse block. These multiscale basis functions are coupled via the symmetric interior penalty discontinuous Galerkin method which provides a block diagonal mass matrix and, consequently, results in fast computations in an explicit time discretization. Our methods' stability and spectral convergence are rigorously analyzed. Numerical examples are presented to show our methods' performance. We also test oversampling strategies. In particular, we discuss how the modes from different snapshot spaces can affect the proposed methods' accuracy.en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/130926675en
dc.rightsArchived with thanks to Multiscale Modeling & Simulationen
dc.subjectwave propagationen
dc.subjectheterogeneous mediaen
dc.subjectmultiscale finite element methoden
dc.subjectModel reductionen
dc.titleGeneralized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Mediaen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalMultiscale Modeling & Simulationen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionDepartment of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR.en
dc.contributor.institutionDepartment of Mathematics, Texas A&M University, College Station, TX 77843en
kaust.authorEfendiev, Yalchin R.en
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