  In this paper we present a new multiscale discontinuous Petrov--Galerkin
method (MsDPGM) for multiscale elliptic problems. This method utilizes the
classical oversampling multiscale basis in the framework of Petrov--Galerkin
version of discontinuous Galerkin finite element method, allowing us to better
cope with multiscale features in the solution. The introduced MsDPGM takes
advantages of the multiscale Petrov--Galerkin method (MsPGM) and discontinuous
Galerkin method (DGM), which can eliminate the resonance error completely, and
can decrease the computational complexity, allowing for more efficient solution
algorithms. Upon the $H^2$ norm error estimate between the multiscale solution
and the homogenized solution with the first order corrector, we give a detailed
multiscale convergence analysis under the assumption that the oscillating
coefficient is periodic. We also investigate the corresponding multiscale
discontinuous finite element method (MsDFEM) which coupling the classical
oversampling multiscale basis with DGM since it has not been studied detailedly
in both aspects of error analysis and numerical tests in the literature.
Numerical experiments are carried out for the multiscale elliptic problems with
periodic and randomly generated log-normal coefficients to demonstrate the
proposed method.

Numerical Analysis