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A residual driven multiscale method for Darcy's flow in perforated domains

Wei Xie, Shubin Fu, Yin Yang, Yunqing Huang

Abstract

In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing complexity by focusing on the dominant pressure variable. Our method is developed within the Generalized Multiscale Finite Element Method (GMsFEM) framework. For each coarse block, we construct offline basis functions from local spectral problems that capture key geometric and physical features. Online basis functions are then adaptively enriched using residuals, allowing the method to incorporate global effects such as source terms and boundary conditions, thereby improving accuracy. We provide detailed error analysis demonstrating how the offline and online spaces contribute to the accuracy and efficiency of the solution. Numerical experiments confirm the method's effectiveness, showing substantial reductions in computational cost while maintaining high accuracy, particularly through adaptive online enrichment. These results highlight the method's potential for efficient and accurate simulation of Darcy flow in complex, heterogeneous perforated domains.

A residual driven multiscale method for Darcy's flow in perforated domains

Abstract

In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing complexity by focusing on the dominant pressure variable. Our method is developed within the Generalized Multiscale Finite Element Method (GMsFEM) framework. For each coarse block, we construct offline basis functions from local spectral problems that capture key geometric and physical features. Online basis functions are then adaptively enriched using residuals, allowing the method to incorporate global effects such as source terms and boundary conditions, thereby improving accuracy. We provide detailed error analysis demonstrating how the offline and online spaces contribute to the accuracy and efficiency of the solution. Numerical experiments confirm the method's effectiveness, showing substantial reductions in computational cost while maintaining high accuracy, particularly through adaptive online enrichment. These results highlight the method's potential for efficient and accurate simulation of Darcy flow in complex, heterogeneous perforated domains.

Paper Structure

This paper contains 16 sections, 4 theorems, 55 equations, 10 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setup
  4. Fine-grid discretization
  5. Offline basis functions
  6. Online basis functions
  7. Analysis
  8. Error analysis of the offline stage
  9. Remark.
  10. Error analysis of the online stage
  11. Remark.
  12. Numerical results
  13. Effect of offline parameters
  14. Offline bases vs. Online bases
  15. Effect of Offline Basis Number and Adaptive Parameter theta
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathbf{u}_h \in \mathbf{V}_h$ be the fine-grid solution of eq:weak_rt_fine, and let $\widehat{\mathbf{u}} \in \mathbf{V}_h$ be the solution of eq:pde_mixed_K_L2f. Then, we have where $\kappa_{\min} = \min_{x \in \Omega^{\epsilon}} \kappa$.

Figures (10)

  • Figure 1: Illustration of a perforated domain $\Omega^{\epsilon}$.
  • Figure 2: Two fine-grid elements, $w_1$ and $w_2$, sharing a common edge $e$.
  • Figure 3: Illustration of the coarse grid and the fine-grid. For perforated regions, fine-grid cells that lie entirely inside the perforations are excluded from the computational domain, while the remaining cells adjacent to the perforation boundaries are treated with appropriate boundary conditions.
  • Figure 4: Three different perforated media. The fine grid consists of 511,756 cells for Model 1; 264,863 for Model 2; and 338,927 for Model 3.
  • Figure 5: Errors with offline and online enrichment for three models: (a, b) Model 1; (c, d) Model 2; (e, f) Model 3.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof