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    Residual-driven online generalized multiscale finite element methods

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    Type
    Article
    Authors
    Chung, Eric T.
    Efendiev, Yalchin R. cc
    Leung, Wing Tat
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Numerical Porous Media SRI Center (NumPor)
    Date
    2015-09-08
    Online Publication Date
    2015-09-08
    Print Publication Date
    2015-12
    Permanent link to this record
    http://hdl.handle.net/10754/577236
    
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    Abstract
    The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces.
    Citation
    Residual-driven online generalized multiscale finite element methods 2015 Journal of Computational Physics
    Publisher
    Elsevier BV
    Journal
    Journal of Computational Physics
    DOI
    10.1016/j.jcp.2015.07.068
    arXiv
    1501.04565
    Additional Links
    http://linkinghub.elsevier.com/retrieve/pii/S0021999115005744
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.jcp.2015.07.068
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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