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CEM-GMsFEM for Poisson equations in heterogeneous perforated domains

Wei Xie, Yin Yang, Eric Chung, Yunqing Huang

TL;DR

We address the Poisson equation in heterogeneous perforated domains by developing a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). The method builds an auxiliary space $V_{\text{aux}}$ from local eigenproblems and then constructs multiscale basis functions on oversampled regions via energy minimization, in two variants: constrained and relaxed. Convergence analysis shows that the basis-function decay and the overall error depend on local eigenvalues, with oversampling depth tunable through these spectral quantities, enabling robust, contrast-insensitive performance. Numerical experiments on two perforated-domain cases demonstrate linear convergence in the coarse mesh size $H$ and illustrate the practical choice of oversampling depth and the impact of eigenfunction enrichment on accuracy.

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.

CEM-GMsFEM for Poisson equations in heterogeneous perforated domains

TL;DR

We address the Poisson equation in heterogeneous perforated domains by developing a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). The method builds an auxiliary space from local eigenproblems and then constructs multiscale basis functions on oversampled regions via energy minimization, in two variants: constrained and relaxed. Convergence analysis shows that the basis-function decay and the overall error depend on local eigenvalues, with oversampling depth tunable through these spectral quantities, enabling robust, contrast-insensitive performance. Numerical experiments on two perforated-domain cases demonstrate linear convergence in the coarse mesh size and illustrate the practical choice of oversampling depth and the impact of eigenfunction enrichment on accuracy.

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.
Paper Structure (15 sections, 9 theorems, 105 equations, 12 figures, 2 tables)

This paper contains 15 sections, 9 theorems, 105 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The construction of the CEM-GMsFEM basis functions
  4. Auxiliary space
  5. Multiscale space
  6. Convergence analysis
  7. Constraint Version
  8. Step 1:
  9. Step 2:
  10. Step 3:
  11. Relaxed Version
  12. Numerical results
  13. Case 1
  14. Case 2
  15. Conclusions

Key Result

Lemma 1

Let $k_i \ge 2$ be an integer and define $W \coloneqq \{ v \in V ~|~ v|_{K_i} \notin V_{\textnormal{aux}}^i \}$. Then, the following inequalities hold

Figures (12)

  • Figure 1: Illustration of a perforated domain.
  • Figure 2: Illustration of the coarse grid (Red), the fine-grid (Blue) and the oversampling domain (Green).
  • Figure 3: Two heterogenous perforated media used in the simulations. The left media will be used for case 1, while the others will be used for case 2.
  • Figure 4: Source term of case 1.
  • Figure 5: Relative L2 error (Left) and Relative h1 error (Right) of Constraint version of CEM-GMsFEM. In here, we consider case 1, the source term and domain are depicted in \ref{['pic:case1_f']}.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • Theorem 1
  • proof
  • ...and 5 more