  In this paper, we discuss the application of Generalized Multiscale Finite
Element Method (GMsFEM) to elasticity equation in heterogeneous media. Our
applications are motivated by elastic wave propagation in subsurface where the
subsurface properties can be highly heterogeneous and have high contrast. We
present the construction of main ingredients for GMsFEM such as the snapshot
space and offline spaces. The latter is constructed using local spectral
decomposition in the snapshot space. The spectral decomposition is based on the
analysis which is provided in the paper. We consider both continuous Galerkin
and discontinuous Galerkin coupling of basis functions. Both approaches have
their cons and pros. Continuous Galerkin methods allow avoiding penalty
parameters though they involve partition of unity functions which can alter the
properties of multiscale basis functions. On the other hand, discontinuous
Galerkin techniques allow gluing multiscale basis functions without any
modifications. Because basis functions are constructed independently from each
other, this approach provides an advantage. We discuss the use of oversampling
techniques that use snapshots in larger regions to construct the offline space.
We provide numerical results to show that one can accurately approximate the
solution using reduced number of degrees of freedom.
