In this paper, we develop efficient multiscale methods for flows in heterogeneous media. We use the generalized multiscale finite element (GMsFEM) framework. GMsFEM approximates the solution space locally using a few multiscale basis functions. This approximation selects an appropriate snapshot space and a local spectral decomposition, e.g., the use of oversampled regions, in order to achieve an efficient model reduction. However, the successful construction of snapshot spaces may be costly if too many local problems need to be solved in order to obtain these spaces. We use a moderate quantity of local solutions (or snapshot vectors) with random boundary conditions on oversampled regions with zero forcing to deliver an efficient methodology. Motivated by the randomized algorithm presented in [P. G. Martinsson, V. Rokhlin, and M. Tygert, A Randomized Algorithm for the approximation of Matrices, YALEU/DCS/TR-1361, Yale University, 2006], we consider a snapshot space which consists of harmonic extensions of random boundary conditions defined in a domain larger than the target region. Furthermore, we perform an eigenvalue decomposition in this small space. We study the application of randomized sampling for GMsFEM in conjunction with adaptivity, where local multiscale spaces are adaptively enriched. Convergence analysis is provided. We present representative numerical results to validate the method proposed.
|Original language||English (US)|
|Number of pages||20|
|Journal||Multiscale Modeling & Simulation|
|State||Published - Mar 23 2016|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: Yalchin Efendiev would like to thank the partial support
from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award DE-FG02-
13ER26165 and the DoD Army ARO Project.