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.

翻译：本文提出了一种新颖的多尺度模型降阶策略，专门用于求解非均匀多孔域中的泊松方程。该复杂问题的数值模拟受制于其多尺度特性，需采用极细网格才能充分捕捉所有相关细节。为克服多孔结构多尺度特性带来的挑战，我们引入了一个基于约束能量最小化广义多尺度有限元方法（CEM-GMsFEM）构建的粗空间。该空间通过在一系列包含非均匀性局部化信息的特征空间上求解局部能量最小化问题来构造基函数。理论分析表明，超采样层数依赖于局部特征值，从而也与局部几何结构相关。此外，我们提供了数值算例以验证所提方案的有效性。