  Motivated by applications to numerical simulation of flows in highly
heterogeneous porous media, we develop multiscale finite element methods for
second order elliptic equations. We discuss a multiscale model reduction
technique in the framework of the discontinuous Galerkin finite element method.
We propose three different finite element spaces on the coarse mesh.
  The first space is based on a local eigenvalue problem that uses a weighted
$L_2-$norm for computing the "mass" matrix. The second space is generated by
amending the eigenvalue problem of the first case with a term related to the
penalty. The third choice is based on generation of a large space of snapshots
and subsequent selection of a subspace of a reduced dimension.
  The approximation with these spaces is based on the discontinuous Galerkin
finite element method framework. We investigate the stability and derive error
estimates for the methods and further experimentally study their performance on
a representative number of numerical examples.
