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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1461-1483 |
| Number of pages | 23 |
| Journal | Multiscale Modeling & Simulation |
| Volume | 8 |
| Issue number | 4 |
| DOIs | |
| State | Published - Jan 2010 |
| Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: 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 ([email protected], [email protected]). The second author's research was partially supported by the DOE and NSF (DMS 0934837, DMS 0902552, and DMS 0811180).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.