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Finite element method coupled with multiscale finite element method for the non-stationary Stokes-Darcy model

Yachen Hong, Wenhan Zhang, Lina Zhao, Haibiao Zheng

TL;DR

This work introduces a multiscale finite element method integrated with an implicit–explicit time-stepping scheme (MsFEM–ImEx) to solve the non-stationary Stokes–Darcy model with multiscale, periodic Darcy permeability. The Darcy region is treated with locally computed multiscale basis functions in offline steps, while the coupled Stokes–Darcy system is advanced on a coarse grid online, yielding substantial computational savings without sacrificing accuracy. The authors establish stability under a timestep constraint and derive an error bound showing dependence on the timestep, the multiscale parameter $\epsilon$, and the coarse mesh size $h$, including the $\epsilon/h$ interaction term. Numerical experiments in FreeFEM++ corroborate the theoretical results, showing improved accuracy over standard FEM on coarse grids and clear advantage of the MsFEM–ImEx scheme in handling multiscale permeability, with clear temporal first-order convergence. This approach enables efficient and reliable simulation of surface–subsurface flows in heterogeneous media with multiscale features, with potential impact on environmental and engineering applications.

Abstract

In this paper, we combine the multiscale flnite element method to propose an algorithm for solving the non-stationary Stokes-Darcy model, where the permeability coefflcient in the Darcy region exhibits multiscale characteristics. Our algorithm involves two steps: first, conducting the parallel computation of multiscale basis functions in the Darcy region. Second, based on these multiscale basis functions, we employ an implicitexplicit scheme to solve the Stokes-Darcy equations. One signiflcant feature of the algorithm is that it solves problems on relatively coarse grids, thus signiflcantly reducing computational costs. Moreover, under the same coarse grid size, it exhibits higher accuracy compared to standard flnite element method. Under the assumption that the permeability coefflcient is periodic and independent of time, this paper demonstrates the stability and convergence of the algorithm. Finally, the rationality and effectiveness of the algorithm are verifled through three numerical experiments, with experimental results consistent with theoretical analysis.

Finite element method coupled with multiscale finite element method for the non-stationary Stokes-Darcy model

TL;DR

This work introduces a multiscale finite element method integrated with an implicit–explicit time-stepping scheme (MsFEM–ImEx) to solve the non-stationary Stokes–Darcy model with multiscale, periodic Darcy permeability. The Darcy region is treated with locally computed multiscale basis functions in offline steps, while the coupled Stokes–Darcy system is advanced on a coarse grid online, yielding substantial computational savings without sacrificing accuracy. The authors establish stability under a timestep constraint and derive an error bound showing dependence on the timestep, the multiscale parameter , and the coarse mesh size , including the interaction term. Numerical experiments in FreeFEM++ corroborate the theoretical results, showing improved accuracy over standard FEM on coarse grids and clear advantage of the MsFEM–ImEx scheme in handling multiscale permeability, with clear temporal first-order convergence. This approach enables efficient and reliable simulation of surface–subsurface flows in heterogeneous media with multiscale features, with potential impact on environmental and engineering applications.

Abstract

In this paper, we combine the multiscale flnite element method to propose an algorithm for solving the non-stationary Stokes-Darcy model, where the permeability coefflcient in the Darcy region exhibits multiscale characteristics. Our algorithm involves two steps: first, conducting the parallel computation of multiscale basis functions in the Darcy region. Second, based on these multiscale basis functions, we employ an implicitexplicit scheme to solve the Stokes-Darcy equations. One signiflcant feature of the algorithm is that it solves problems on relatively coarse grids, thus signiflcantly reducing computational costs. Moreover, under the same coarse grid size, it exhibits higher accuracy compared to standard flnite element method. Under the assumption that the permeability coefflcient is periodic and independent of time, this paper demonstrates the stability and convergence of the algorithm. Finally, the rationality and effectiveness of the algorithm are verifled through three numerical experiments, with experimental results consistent with theoretical analysis.
Paper Structure (13 sections, 2 theorems, 67 equations, 2 figures, 8 tables, 1 algorithm)

This paper contains 13 sections, 2 theorems, 67 equations, 2 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Coupled Stokes-Darcy model
  3. Problem
  4. Weak formulation of Stokes-Darcy model
  5. Time and space discretization
  6. The multiscale finite element method for the Stokes-Darcy model
  7. Numerical algorithm
  8. Stability analysis
  9. Error estimates
  10. Numerical experiments
  11. Implementation
  12. Numerical results
  13. Conclusion

Key Result

Theorem 3.1

Under the assumption we have the following stability result: where $C(T)$ is a positive constant which depends on the final time $T$. The constants $C_{1}, C_{t}, \widetilde{C}_{t}, {C}_{p}$ and $\widetilde{C}_{p}$ are respectively associated with Korn's inequality, the trace inequality, and the $\text{Poincaré}$ inequality.

Figures (2)

  • Figure 1: A global domain $\Omega$ consisting of a fluid flow region $\Omega_f$ and a porous media flow region $\Omega_p$ separated by an interface $\Gamma$.
  • Figure 2: For $\epsilon=0.008$, the pressure distribution at $t=1.0$ for the reference solution(a), the pressure distribution at $t=1.0$, $h=1/64$ with MsFEM-ImEx scheme(b).

Theorems & Definitions (9)

  • Remark 1
  • Theorem 3.1: Stability
  • proof
  • Remark 2
  • Theorem 3.2: Error
  • proof
  • Example 1
  • Example 2
  • Example 3