Numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas. However, numerical solutions on the fine grid are often prohibitively expensive, and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid. In this paper, we propose an energy minimization based multiscale model reduction approach in the discontinuous Galerkin discretization setting. The main idea of the method is to extract the non-decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems, and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems. The multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous Galerkin formulation. The minimal degree of freedom in representing highcontrast features is achieved through the design of local spectral problems, which provides the most compressed local multiscale space. We analyze the method for solving Darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast, provided that the oversampling size is appropriately chosen. Numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high-contrast heterogeneous media.heterogeneities in the medium properties. In numerical homogenization approaches, effective properties are computed and the global problem is formulated and solved on the coarse grid. However, these approaches are limited to the cases when the medium properties possess scale separation. On the other hand, multiscale methods construct of multiscale basis functions which are responsible for capturing the local oscillatory effects of the solution. Once the multiscale basis functions, coarse-scale equations are formulated. Moreover, fine-scale information can be recovered by the coarse-scale coefficients and mutliscale basis functions.Many existing mutliscale methods, such as MsFEM, VMS and HMM, construct one basis function per local coarse region to handle the effects of local heterogeneities. However, for more complex multiscale problems, each local coarse region contains several high-conductivity regions and multiple multiscale basis functions are required to represent the local solution space. GMsFEM is developed to allow systematic enrichment of the coarse-scale space with fine-scale information and identify the underlying low-dimensional local structures for solution representation. The main idea of GMsFEM is to extract local dominant modes by carefully designed local spectral problems in coarse regions, and the convergence of the GMsFEM is related to eigenvalue decay of local spectral problems. For a more detailed discussion on GMsFEM, we refer the readers to [21,18,20,15,11,8,25,6,36,38] and the references therein. Our method developed in this work is mo...
