Multiscale finite element method for Stokes-Darcy model

TL;DR

This paper develops a multiscale finite element method for efficiently solving steady-state Stokes-Darcy problems in highly heterogeneous porous media, combining offline basis construction with an online Robin-Robin algorithm and validating through numerical tests.

Contribution

It introduces a novel multiscale finite element algorithm for Stokes-Darcy models with error analysis and parallel offline basis construction for heterogeneous media.

Findings

Effective coarse-grid simulation in heterogeneous media

Error bounds established for the proposed method

Numerical tests confirm the method's accuracy and efficiency

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.