Multiscale finite element method for Stokes-Darcy model

TL;DR

The paper addresses steady Stokes-Darcy flow with Beavers-Joseph-Saffman interface in multiscale porous media. It develops a parallelizable MsFEM framework with offline Darcy bases and a Robin-Robin online coupling to the Stokes region, supported by a priori $L^2$ and $H^1$ error bounds under periodic coefficients. Theoretical analysis leverages homogenization and projection techniques to quantify errors, while numerical experiments on highly heterogeneous media confirm expected convergence behavior and demonstrate the efficiency gains from offline parallel basis construction. The work provides a practical, scalable approach for accurate coarse-grid simulations of multiscale Stokes-Darcy systems with BJS interfaces, enabling robust predictions in complex porous media applications.

Abstract

This paper explores the application of the multiscale finite element method (MsFEM) to address steady-state Stokes-Darcy problems with BJS interface conditions in highly heterogeneous porous media. We assume the existence of multiscale features in the Darcy region and propose an algorithm for the multiscale Stokes-Darcy model. During the offline phase, we employ MsFEM to construct permeability-dependent offline bases for efficient coarse-grid simulation, with this process conducted in parallel to enhance its efficiency. In the online phase, we use the Robin-Robin algorithm to derive the model's solution. Subsequently, we conduct error analysis based on $L^2$ and $H^1$ norms, assuming certain periodic coefficients in the Darcy region. To validate our approach, we present extensive numerical tests on highly heterogeneous media, illustrating the results of the error analysis.

Figures (5)

• Figure 1:
• Figure 2: Sketch of MsFEM basis function design in 2D (a), Example of MsFEM basis function (b) and P1 piecewise function (c)
• Figure 4: Parallel Efficient Generation of Base Functions
• Figure 5: $\mathbb{K}_{\epsilon}$ in Example 1
• Figure 6: $\mathbb{K}_{\epsilon}$ in Example 2

Theorems & Definitions (11)

• Theorem 2.1: Proofs in wilbrandt2019stokes
• Remark 2.1
• Theorem 2.2: Proofs in wilbrandt2019stokes
• Definition 2.1
• Theorem 2.3
• Theorem 4.1: $H^1$ norm
• proof
• Remark 4.1
• Theorem 4.2: L2 norm
• proof