Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Handle URI:
http://hdl.handle.net/10754/598008
Title:
Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media
Authors:
Galvis, Juan; Efendiev, Yalchin
Abstract:
In this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent of the contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper. © 2010 Society for Industrial and Applied Mathematics.
Citation:
Galvis J, Efendiev Y (2010) Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media. Multiscale Model Simul 8: 1461–1483. Available: http://dx.doi.org/10.1137/090751190.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
Multiscale Modeling & Simulation
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Jan-2010
DOI:
10.1137/090751190
Type:
Article
ISSN:
1540-3459; 1540-3467
Sponsors:
Received by the editors March 1, 2009; accepted for publication (in revised form) May 10, 2010; published electronically August 5, 2010. This work was partially supported by award KUS-C1-016-04 from King Abdullah University of Science and Technology (KAUST).Department of Mathematics, Texas A&M University, College Station, TX 77843 (jugal@math.tamu.edu, efendiev@math.tamu.edu). The second author's research was partially supported by the DOE and NSF (DMS 0934837, DMS 0902552, and DMS 0811180).
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Full metadata record

DC FieldValue Language
dc.contributor.authorGalvis, Juanen
dc.contributor.authorEfendiev, Yalchinen
dc.date.accessioned2016-02-25T13:10:52Zen
dc.date.available2016-02-25T13:10:52Zen
dc.date.issued2010-01en
dc.identifier.citationGalvis J, Efendiev Y (2010) Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media. Multiscale Model Simul 8: 1461–1483. Available: http://dx.doi.org/10.1137/090751190.en
dc.identifier.issn1540-3459en
dc.identifier.issn1540-3467en
dc.identifier.doi10.1137/090751190en
dc.identifier.urihttp://hdl.handle.net/10754/598008en
dc.description.abstractIn this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent of the contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper. © 2010 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipReceived by the editors March 1, 2009; accepted for publication (in revised form) May 10, 2010; published electronically August 5, 2010. This work was partially supported by award KUS-C1-016-04 from King Abdullah University of Science and Technology (KAUST).Department of Mathematics, Texas A&M University, College Station, TX 77843 (jugal@math.tamu.edu, efendiev@math.tamu.edu). The second author's research was partially supported by the DOE and NSF (DMS 0934837, DMS 0902552, and DMS 0811180).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectCoarse spacesen
dc.subjectDomain decompositionen
dc.subjectHigh-contrast elliptic problemsen
dc.subjectMultiscale problemsen
dc.subjectSpectral constructionsen
dc.titleDomain Decomposition Preconditioners for Multiscale Flows in High-Contrast Mediaen
dc.typeArticleen
dc.identifier.journalMultiscale Modeling & Simulationen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
kaust.grant.numberKUS-C1-016-04en
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