Convergence of the CEM-GMsFEM for Stokes flows in heterogeneous perforated domains

TL;DR

This paper introduces a multiscale finite element method tailored for accurately simulating incompressible Stokes flows in complex perforated domains, emphasizing divergence-free basis functions and spectral convergence.

Contribution

The paper develops a systematic approach to construct divergence-free multiscale basis functions for Stokes flows in heterogeneous perforated domains using CEM-GMsFEM, with proven exponential decay and spectral convergence.

Findings

Basis functions exhibit exponential decay outside local regions.

Spectral convergence with error bounds related to coarse mesh size.

Method effectively handles heterogeneities in perforated domains.

Abstract

In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible and systematical approach to construct crucial divergence-free multiscale basis functions for approximating the displacement field. These basis functions are constructed by solving a class of local energy minimization problems over the eigenspaces that contain local information on the heterogeneities. These multiscale basis functions are shown to have the property of exponential decay outside the corresponding local oversampling regions. By adapting the technique of oversampling, the spectral convergence of the method with error bounds related to the coarse mesh size is proved.