Generalized Multiscale Finite Element Methods for problems in perforated heterogeneous domains

TL;DR

This paper develops a generalized multiscale finite element method (GMsFEM) for efficiently solving complex physical problems in perforated, heterogeneous domains with multiple scales and no clear scale separation.

Contribution

It introduces a GMsFEM-based approach that constructs multiscale basis functions using snapshot spaces and spectral problems, enabling accurate solutions with few basis functions per coarse block.

Findings

Accurately approximates solutions with few basis functions

Applicable to Laplace, elasticity, and Stokes equations in perforated domains

Effective in domains with many small inclusions

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 (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.