Coupling of multiscale and multi-continuum approaches
The Chinese University of Hong Kong（香港中文大学）；Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；Institute of Mathematics and Informatics（数学与信息学研究所）
Eric T. Chung, Yalchin Efendiev, Tat Leung, Maria Vasilyeva
Simulating complex processes in fractured media requires some type of model reduction. Well-known approaches include multi-continuum techniques, which have been commonly used in approximating subgrid effects for flow and transport in fractured media. Our goal in this paper is to (1) show a relation between multi-continuum approaches and Generalized Multiscale Finite Element Method (GMsFEM) and (2) to discuss coupling these approaches for solving problems in complex multiscale fractured media. The GMsFEM, a systematic approach, constructs multiscale basis functions via local spectral decomposition in pre-computed snapshot spaces. We show that GMsFEM can automatically identify separate fracture networks via local spectral problems. We discuss the relation between these basis functions and continuums in multi-continuum methods. The GMsFEM can automatically detect each continuum and represent the interaction between the continuum and its surrounding (matrix). For problems with simplified fracture networks, we propose a simplified basis construction with the GMsFEM. This simplified approach is effective when the fracture networks are known and have simplified geometries. We show that this approach can achieve a similar result compared to the results using the GMsFEM with spectral basis functions. Further, we discuss the coupling between the GMsFEM and multi-continuum approaches. In this case, many fractures are resolved while for unresolved fractures, we use a multi-continuum approach with local Representative Volume Element (RVE) information. As a result, the method deals with a system of equations on a coarse grid, where each equation represents one of the continua on the fine grid. We present various basis construction mechanisms and numerical results.
Constraint Energy Minimizing Generalized Multiscale Finite Element Method for dual continuum model
Texas A&M University（德克萨斯农工大学）；The Chinese University of Hong Kong（香港中文大学）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；North-Eastern Federal University（东北联邦大学）
Higher-Order Multiscale Computational Method for Multi-Continuum Problems in Highly Heterogeneous Media
School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC)
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong（香港中文大学）
Generalized Multiscale Finite Element Method. Symmetric Interior Penalty Coupling
Texas A&M University（德克萨斯农工大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；Instituto Nacional de Matemática Pura e Aplicada（巴西纯数学与应用数学研究所）
Noise-balanced multilevel on-the-fly sparse grid surrogates for coupling Monte Carlo models into continuum models with application to heterogeneous catalysis
Fritz Haber Institute of the Max Planck Society（马克斯·普朗克学会弗里茨·哈伯研究所）
