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The pressure-robust weak Galerkin finite element method for Stokes-Darcy problem

Jiwei Jia, Lin Yang, Qilong Zhai

TL;DR

This work addresses solving the coupled Stokes-Darcy system by introducing a pressure-robust weak Galerkin discretization that leverages a divergence-free velocity reconstruction operator $R_T$ mapping to $RT_k(T)$ to enforce $ abladot(R_T({f v}_h))= abla_wdot{f v}_h$. By modifying the test function on the RHS with reconstructed velocities, the scheme decouples velocity errors from pressure and viscosity, achieving optimal convergence in the energy and $L^2$ norms. The authors establish stability, derive error equations, and prove energy-norm and pressure norms converge with rates $|||f e_h||| o C h^k orm{f u}_{k+1}$ and $ orm{oldsymbol{eta}_h} o ext{bounded by } u h^k orm{f u}_{k+1}$. Numerical experiments confirm pressure-robustness and optimal performance across a range of viscosities and pressures. Overall, the method provides reliable, high-accuracy simulations for Stokes-Darcy flows with Beavers-Joseph-Saffman coupling in environmental and engineering contexts.

Abstract

In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function and the pressure function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

The pressure-robust weak Galerkin finite element method for Stokes-Darcy problem

TL;DR

This work addresses solving the coupled Stokes-Darcy system by introducing a pressure-robust weak Galerkin discretization that leverages a divergence-free velocity reconstruction operator mapping to to enforce . By modifying the test function on the RHS with reconstructed velocities, the scheme decouples velocity errors from pressure and viscosity, achieving optimal convergence in the energy and norms. The authors establish stability, derive error equations, and prove energy-norm and pressure norms converge with rates and . Numerical experiments confirm pressure-robustness and optimal performance across a range of viscosities and pressures. Overall, the method provides reliable, high-accuracy simulations for Stokes-Darcy flows with Beavers-Joseph-Saffman coupling in environmental and engineering contexts.

Abstract

In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function and the pressure function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.
Paper Structure (11 sections, 11 theorems, 65 equations, 3 figures, 10 tables, 2 algorithms)

This paper contains 11 sections, 11 theorems, 65 equations, 3 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Weak Galerkin Finite Element Scheme
  3. Weak Galerkin finite element spaces
  4. The numerical scheme
  5. Stability and Well-posedness
  6. Error Analysis
  7. The properties of the velocity reconstruction operator
  8. Error equation
  9. Error Estimates in the Energy Norm
  10. Numerical Results
  11. Conclusion

Key Result

Lemma 3.1

For ${\bf u} \in [H^1(\Omega)]^2$ and $T \in \mathcal{T}_h$, we have the following properties:

Figures (3)

  • Figure 1.1: The Stokes-Darcy Model
  • Figure 6.1: Comparison of errors between Algorithms \ref{['Algorithm1']} and \ref{['Algorithm2']} in the Stokes domain with different $\mu$
  • Figure 6.2: Comparison of errors between Algorithms \ref{['Algorithm1']} and \ref{['Algorithm2']} in the Darcy domain with different $\mu$

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.1
  • Lemma 4.1
  • ...and 6 more