In this talk, we first give a brief introduction to the discontinuous
Galerkin method, which is a finite element method using completely
discontinuous basis functions, for solving hyperbolic conservation
laws and parabolic and elliptic equations. We will then survey the
progress in developing discontinuous Galerkin methods for multiscale
problems, in three different approaches, namely using the
heterogeneous multiscale method (HMM) framework, using domain
decompositions, and using multiscale basis in the discontinuous
Galerkin method. The emphasis is on the last approach.
Numerical results will be shown to demonstrate the effectiveness
of the multiscale discontinuous Galerkin methods.