Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation

Generalized Multiscale Finite Element Methods for problems in perforated heterogeneous domains

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). Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works [14, 18, 17], where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can accurately approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domain; (2) elasticity equation in perforated domain; and (3) Stokes equations in perforated domain. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere

Adaptive Generalized Multiscale Model Reduction Techniques for Problems in Perforated Domains

Multiscale modeling of complex physical phenomena in many areas, including hydrogeology, material science, chemistry and biology, consists of solving problems in highly heterogeneous porous media. In many of these applications, differential equations are formulated in perforated domains which can be considered as the region outside of inclusions or connected bodies of various sizes. Due to complicated geometries of these inclusions, solutions to these problems have multiscale features. Taking into account the uncertainties, one needs to solve these problems extensively many times. Model reduction techniques are significant for problems in perforated domains in order to improve the computational efficiency. There are some existing approaches for model reduction in perforated domains including homogenization, heterogeneous multiscale methods and multiscale finite element methods. These techniques typically consider the case when there is a scale separation or the perforation distribution is periodic, and assume that the solution space can be approximated by the solutions of directional cell problems and the effective equations contain a limited number of effective parameters. For more complicated problems where the effective properties may be richer, we are interested in developing systematic local multiscale model reduction techniques to obtain accurate macroscale representations of the underlying fine-scale problem in highly heterogeneous perforated domains. In this dissertation, based on the framework of Generalized Multiscale Finite Element Method, we develop novel methods and algorithms including (1) development of systematic local model reduction techniques for computing multiscale basis in perforated domains, (2) numerical analysis and exhaustive simulation utilizing the proposed basis functions, (3) design of different applicable global coupling frameworks and (4) applications to various problems with challenging engineering backgrounds. Our proposed methods can significantly advance the computational efficiency and accuracy for multiscale problems in perforated media

Nonlinear nonlocal multicontinua upscaling framework and its applications

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 intended for problems without scale separation and high contrast, which often occur in applications. For linear problems, the local solutions with constraints are used as basis functions. This technique is called Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). GMsFEM identifies macroscopic quantities based on rigorous analysis. In corresponding upscaling methods, the multiscale basis functions are selected such that the degrees of freedom have physical meanings, such as averages of the solution on each continuum. This paper extends the linear concepts to nonlinear problems, where the local problems are nonlinear. The main concept consists of: (1) identifying macroscopic quantities; (2) constructing appropriate oversampled local problems with coarse-grid constraints; (3) formulating macroscopic equations. We consider two types of approaches. In the first approach, the solutions of local problems are used as basis functions (in a linear fashion) to solve nonlinear problems. This approach is simple to implement; however, it lacks the nonlinear interpolation, which we present in our second approach. In this approach, the local solutions are used as a nonlinear forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem. Both approaches are discussed on several examples and applied to single-phase and two-phase flow problems, which are challenging because of convection-dominated nature of the concentration equation