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Generalized Multiscale Finite Element Methods. Oversampling Strategies

In this paper, we propose oversampling strategies in the Generalized Multiscale Finite Element Method (GMsFEM) framework. The GMsFEM, which has been recently introduced in [12], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The main idea of the method consists of (1) the construction of snapshot space, (2) the construction of the offline space, and (3) construction of the online space (the latter for parameter-dependent problems). In [12], it was shown that the GMsFEM provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot, offline, and online spaces. In this paper, we develop oversampling techniques to be used in this context (see [19] where oversampling is introduced for multiscale finite element methods). It is known (see [19]) that the oversampling can improve the accuracy of multiscale methods. In particular, the oversampling technique uses larger regions (larger than the target coarse block) in constructing local basis functions. Our motivation stems from the analysis presented in this paper which show that when using oversampling techniques in the construction of the snapshot space and offline space, GMsFEM will converge independent of small scales and high-contrast under certain assumptions. We consider the use of multiple eigenvalue problem to improve the convergence and discuss their relation to single spectral problems that use oversampled regions. The oversampling procedures proposed in this paper differ from those in [19]. In particular, the oversampling domains are partially used in constructing local spectral problems. We present numerical results and compare various oversampling techniques in order to complement the proposed technique and analysis.