  Numerical simulations of waves in highly heterogeneous media have important
applications, but direct computations are prohibitively expensive. In this
paper, we develop a new generalized multiscale finite element method with the
aim of simulating waves at a much lower cost. Our method is based on a mixed
Galerkin type method with carefully designed basis functions that can capture
various scales in the solution. The basis functions are constructed based on
some local snapshot spaces and local spectral problems defined on them. The
spectral problems give a natural ordering of the basis functions in the
snapshot space and allow systematically enrichment of basis functions. In
addition, by using a staggered coarse mesh, our method is energy conserving and
has block diagonal mass matrix, which are desirable properties for wave
propagation. We will prove that our method has spectral convergence, and
present numerical results to show the performance of the method.
