  In this paper, we propose a general approach called Generalized Multiscale
Finite Element Method (GMsFEM) for performing multiscale simulations for
problems without scale separation over a complex input space. As in multiscale
finite element methods (MsFEMs), the main idea of the proposed approach is to
construct a small dimensional local solution space that can be used to generate
an efficient and accurate approximation to the multiscale solution with a
potentially high dimensional input parameter space. In the proposed approach,
we present a general procedure to construct the offline space that is used for
a systematic enrichment of the coarse solution space in the online stage. The
enrichment in the online stage is performed based on a spectral decomposition
of the offline space. In the online stage, for any input parameter, a
multiscale space is constructed to solve the global problem on a coarse grid.
The online space is constructed via a spectral decomposition of the offline
space and by choosing the eigenvectors corresponding to the largest
eigenvalues. The computational saving is due to the fact that the construction
of the online multiscale space for any input parameter is fast and this space
can be re-used for solving the forward problem with any forcing and boundary
condition. Compared with the other approaches where global snapshots are used,
the local approach that we present in this paper allows us to eliminate
unnecessary degrees of freedom on a coarse-grid level. We present various
examples in the paper and some numerical results to demonstrate the
effectiveness of our method.
