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Optimized Schwarz method for the Stokes-Darcy problem with generalized interface conditions

Paula Strohbeck, Marco Discacciati, Iryna Rybak

TL;DR

The paper addresses solving the steady Stokes–Darcy coupling under generalized Beavers–Joseph-type interface conditions derived from homogenization. It develops an optimized Robin–Robin domain decomposition method and uses Fourier analysis to select optimal interface weights by minimizing the frequency-dependent reduction factor $\tilde{\rho}$ on the curve $α_{ff}α_{pm}=2/\sqrt{κ_{11}κ_{22}}$. Explicit optimal parameters $(α_{ff}^{*}, α_{pm}^{*})$ are derived, and robustness is demonstrated with respect to mesh size, scale separation $ε$, and boundary-layer coefficient, with the permeability $κ$ being the most influential. Numerical experiments on analytical and general filtration problems confirm improved convergence and stability compared to classical coupling strategies, highlighting the method’s practical relevance for environmental, industrial, and medical flow simulations.

Abstract

Due to their wide appearance in environmental settings as well as industrial and medical applications, the Stokes-Darcy problems with different sets of interface conditions establish an active research area in the community of mathematical modelers and computational scientists. For numerical simulation of such coupled problems in applications, robust and efficient computational algorithms are needed. In this work, we consider a generalization of the Beavers-Joseph interface condition recently developed using homogenization and boundary layer theory. This extension is applicable not only for the parallel flows to the fluid-porous interface as its predecessor, but also for arbitrary flow directions. To solve the Stokes-Darcy problem with these generalized interface conditions efficiently, we develop and analyze a Robin-Robin domain decomposition method using Fourier analysis to identify optimal weights in the Robin interface conditions. We study efficiency and robustness of the proposed method and provide numerical simulations which confirm the obtained theoretical results.

Optimized Schwarz method for the Stokes-Darcy problem with generalized interface conditions

TL;DR

The paper addresses solving the steady Stokes–Darcy coupling under generalized Beavers–Joseph-type interface conditions derived from homogenization. It develops an optimized Robin–Robin domain decomposition method and uses Fourier analysis to select optimal interface weights by minimizing the frequency-dependent reduction factor on the curve . Explicit optimal parameters are derived, and robustness is demonstrated with respect to mesh size, scale separation , and boundary-layer coefficient, with the permeability being the most influential. Numerical experiments on analytical and general filtration problems confirm improved convergence and stability compared to classical coupling strategies, highlighting the method’s practical relevance for environmental, industrial, and medical flow simulations.

Abstract

Due to their wide appearance in environmental settings as well as industrial and medical applications, the Stokes-Darcy problems with different sets of interface conditions establish an active research area in the community of mathematical modelers and computational scientists. For numerical simulation of such coupled problems in applications, robust and efficient computational algorithms are needed. In this work, we consider a generalization of the Beavers-Joseph interface condition recently developed using homogenization and boundary layer theory. This extension is applicable not only for the parallel flows to the fluid-porous interface as its predecessor, but also for arbitrary flow directions. To solve the Stokes-Darcy problem with these generalized interface conditions efficiently, we develop and analyze a Robin-Robin domain decomposition method using Fourier analysis to identify optimal weights in the Robin interface conditions. We study efficiency and robustness of the proposed method and provide numerical simulations which confirm the obtained theoretical results.

Paper Structure

This paper contains 12 sections, 2 theorems, 65 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Coupled Stokes--Darcy flow model
  4. Interface conditions
  5. Robin--Robin method
  6. Formulation of the Robin--Robin method
  7. Analysis of the method
  8. Interface system associated with the Robin--Robin method
  9. Numerical simulation results
  10. Test 1 (analytical solution)
  11. Test 2 (general filtration problem)
  12. Discussion

Key Result

Theorem 3.1

The error reduction factor $\rho(\alpha_{\mathrm{ff}},\alpha_{\mathrm{pm}}, k)$ of the Robin--Robin algorithm eq:Stokes-momentum-simplified-1--eq:IC-Darcy is independent of the iteration $m$, and it can be expressed as with

Figures (2)

  • Figure 1: Flow system description (left) and visualization Strohbeck-Eggenweiler-Rybak-23 of the microscale velocity field (right) for the general filtration problem
  • Figure 2: Error reduction factors versus relevant frequencies $k$ for the values of the coefficients corresponding to Case 2 (left), Case 4 (middle) and Case 8 (right) in Table \ref{['tab:results-case-2b']}

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof