Multiscale discontinuous Petrov--Galerkin method for the multiscale elliptic problems

Abstract: 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.