  Numerical modeling of wave propagation in heterogeneous media is important in
many applications. Due to the complex nature, direct numerical simulations on
the fine grid are prohibitively expensive. It is therefore important to develop
efficient and accurate methods that allow the use of coarse grids. In this
paper, we present a multiscale finite element method for wave propagation on a
coarse grid. The proposed method is based on the Generalized Multiscale Finite
Element Method (GMsFEM). To construct multiscale basis functions, we start with
two snapshot spaces in each coarse-grid block where one represents the degrees
of freedom on the boundary and the other represents the degrees of freedom in
the interior. We use local spectral problems to identify important modes in
each snapshot space. These local spectral problems are different from each
other and their formulations are based on the analysis. To our best knowledge,
this is the first time where multiple snapshot spaces and multiple spectral
problems are used and necessary for efficient computations. Using the dominant
modes from local spectral problems, multiscale basis functions are constructed
to represent the solution space locally within each coarse block. These
multiscale basis functions are coupled via the symmetric interior penalty
discontinuous Galerkin method which provides a block diagonal mass matrix, and,
consequently, results in fast computations in an explicit time discretiza-
tion. Our methods' stability and spectral convergence are rigorously analyzed.
Numerical examples are presented to show our methods' performance. We also test
oversampling strategies. In particular, we discuss how the modes from different
snapshot spaces can affect the proposed methods' accuracy.
