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.

Keywords:

• perforated domain
• multiscale finite element method
• model reduction
• Laplace equation
• Stokes equations
• elasticity equation

AMS Subject Classifications:

• 65M60
• 74S05
• 76M10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

YE’s work is partially supported by the US Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project and NSF (DMS 0934837 and DMS 0811180). Eric Chung’s research is partially supported by Hong Kong RGC General Research Fund (Project: 400813) and CUHK Faculty of Science Research Incentive Fund 2015-16.