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

TL;DR

This paper introduces a multiscale pressure-focused method for simulating Darcy flow in complex perforated domains, significantly reducing computational costs while maintaining high accuracy through adaptive online enrichment.

Contribution

The paper develops a residual-driven multiscale method within GMsFEM that reformulates Darcy flow to focus on pressure, incorporating adaptive basis enrichment for improved efficiency and accuracy.

Findings

Substantial computational cost reduction achieved.

High accuracy maintained with online basis enrichment.

Effective handling of complex geometries demonstrated.

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.