Online Mixed Multiscale Finite Element Method with Oversampling and Its Applications In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an extension of the online mixed generalized multiscale method (Chan et al. in Numer Math Theory Methods Appl 9(4):497xe2x80x93527, 2016). The multiscale online basis functions are computed by solving a Neumann problem in an oversampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method. Extensive numerical experiments are presented to demonstrate the performance of our methods for both steadystate flow, and twophase flow and transport problems.

Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media Abstract   In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for poroelasticity problem in complex heterogeneous media. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach for two and threedimensional formulations. The coarse grid approximation is constructed using the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions for displacement and pressures in multicontinuum media. We present numerical results for the two and three dimensional model problems in heterogeneous fractured porous media. We investigate relative errors between reference fine grid solution and presented coarse grid approximation using GMsFEM with different numbers of multiscale basis functions. Our results indicate that the proposed method is able to give accurate solutions with few degrees of freedoms.