  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.
