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    Randomized oversampling for generalized multiscale finite element methods

    246008_246008.pdf (455.1Kb)
    Access Status
    Open access
    Authors
    Calo, Victor
    Efendiev, Y.
    Galvis, J.
    Li, G.
    Date
    2016
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Calo, V. and Efendiev, Y. and Galvis, J. and Li, G. 2016. Randomized oversampling for generalized multiscale finite element methods. Multiscale Modeling and Simulation. 14 (1): pp. 482-501.
    Source Title
    Multiscale Modeling and Simulation
    DOI
    10.1137/140988826
    ISSN
    1540-3459
    School
    Department of Applied Geology
    Remarks

    Copyright © 2016 Society for Industrial and Applied Mathematics

    URI
    http://hdl.handle.net/20.500.11937/6307
    Collection
    • Curtin Research Publications
    Abstract

    In this paper, we develop efficient multiscale methods for ows in heterogeneous media. We use the generalized multiscale finite element (GMsFEM) framework. GMsFEM approxi- mates 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 random- ized 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.

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