CEM-GMsFEM for Poisson equations in heterogeneous perforated domains

TL;DR

This paper introduces a multiscale reduction method called CEM-GMsFEM for efficiently solving Poisson equations in complex perforated domains with heterogeneous features, reducing computational costs while maintaining accuracy.

Contribution

The paper develops a new multiscale finite element method tailored for perforated domains, incorporating local energy minimization and eigenvalue-based oversampling strategies.

Findings

The method effectively captures multiscale features with fewer degrees of freedom.

Numerical examples confirm the accuracy and efficiency of the proposed scheme.

Oversampling layers are shown to depend on local eigenvalues and geometry.

Abstract

In this paper, we propose a novel multiscale model reduction strategy tailored to address the Poisson equation within heterogeneous perforated domains. The numerical simulation of this intricate problem is impeded by its multiscale characteristics, necessitating an exceptionally fine mesh to adequately capture all relevant details. To overcome the challenges inherent in the multiscale nature of the perforations, we introduce a coarse space constructed using the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). This involves constructing basis functions through a sequence of local energy minimization problems over eigenspaces containing localized information pertaining to the heterogeneities. Through our analysis, we demonstrate that the oversampling layers depend on the local eigenvalues, thereby implicating the local geometry as well. Additionally, we provide numerical examples to illustrate the efficacy of the proposed scheme.