  In this paper, we propose a domain decomposition method for multiscale second
order elliptic partial differential equations with highly varying coefficients.
The method is based on a discontinuous Galerkin formulation. We present both a
nonoverlapping and an overlapping version of the method. We prove that the
condition number bound of the preconditioned algebraic system in either case
can be made independent of the coefficients under certain assumptions. Also, in
our analysis, we do not need to assume that the coefficients are continuous
across the coarse grid boundaries. The analysis and the condition number bounds
are new, and contribute towards further extension of the theory for the
discontinuous Galerkin discretization for multiscale problems.
