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In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the…
In this paper, we discuss multiscale methods for nonlinear problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are…
The oversampling multiscale finite element method (MsFEM) is one of the most popular methods for simulating composite materials and flows in porous media which may have many scales. But the method may be inapplicable or inefficient in some…
In this paper, we examine the effectiveness of classic multiscale finite element method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixed Dirichlet-Neumann, Robin and hemivariational inequality boundary problems. Constructing so-called…
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…
In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and…
In this paper, we study the development of efficient multiscale methods for flows in heterogeneous media. Our approach uses the Generalized Multiscale Finite Element (GMsFEM) framework. The main idea of GMsFEM is to approximate the solution…
Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales (see Figure 1 for the illustration of a perforated domain).…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite…
In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational…
For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property…
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…
Numerical simulations of waves in highly heterogeneous media have important applications, but direct computations are prohibitively expensive. In this paper, we develop a new generalized multiscale finite element method with the aim of…
In this paper, we propose a local-global multiscale method for highly heterogeneous stochastic groundwater flow problems under the framework of reduced basis method and the generalized multiscale finite element method (GMsFEM). Due to…
This paper presents a high-accuracy higher-order multiscale method for solving multi-continuum problems in in highly heterogeneous media. First, microscopic unit cell functions are defined, leading to the derivation of macroscopic…
In this paper, we consider the numerical solution of poroelasticity problems that are of Biot type and develop a general algorithm for solving coupled systems. We discuss the challenges associated with mechanics and flow problems in…
In this paper, we propose a randomized generalized multiscale finite element method (Randomized GMsFEM) for flow problems with parameterized inputs and high-contrast heterogeneous media. The method employs a data-driven predictor to…
This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices,…
In this paper, we propose a deep-learning-based approach to a class of multiscale problems. THe Generalized Multiscale Finite Element Method (GMsFEM) has been proven successful as a model reduction technique of flow problems in…