Generalized multiscale finite element methods for problems in perforated heterogeneous domains

Abstract

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales. 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, 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 approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domains; (2) elasticity equation in perforated domains; and (3) Stokes equations in perforated domains. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere.

A Three-Level Multi-Continua Upscaling Method for Flow Problems in Fractured Porous Media

“…where the matrix A is the restriction of the matrixĀ in the coarse neighborhood ω i and the matrixĀ is the matrix resulting from the NLMC method (5). Moreover, the matrix S is defined as follows:…”

Dynamic data-driven Bayesian GMsFEM

“…In this section, we will discuss the problem settings and the key ingredients of Generalized Multiscale Finite Element Methods (GMsFEM) [18,31,32,19,20,7,14,15,13,12,10]. Several approaches for multiscale model reduction by GMsFEM have been proposed for parabolic equation, and we present a unified discussion of GMsFEM in this section.…”

Multiscale stabilization for convection–diffusion equations with heterogeneous velocity and diffusion coefficients

“…There are in literature a wide range of numerical schemes for this problem that are based on constructions of special basis functions on coarse grids [29,56,3,36,32,37,41,40,31,33,38,16,30,12,13,44,34]. These methods include the Multiscale Finite Element Methods (MsFEM) [36,32,37,38,43,2], the Variational Multiscale Methods [48,46,45,6,52,9,47,23,5,1,50] and the Generalized Multiscale Finite Element Method (GMsFEM) [32,42,43,34,35,10,21,17,11,22,19,20]. When the above approaches are used to solve multiscale convection-dominated diffusion problems with a high Peclet number, besides finding a reduced approximate solution space, one needs to stabilize the system to avoid large errors …”