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Optimal analysis of penalized lowest-order mixed FEMs for the Stokes-Darcy model

Luling Cao, Weiwei Sun

TL;DR

This work addresses the dynamic Stokes–Darcy coupling with Beavers–Joseph–Saffman interface conditions by introducing a decoupled, fully-mixed FEM that uses non-uniform lowest-order approximations (MINI for Stokes and RT0–DG0 for Darcy) and a Nitsche-type penalty for the interface. The authors prove true optimal spatial convergence rates for the coupled system, showing $O(h^2)$ accuracy for the Stokes velocity in $L^2$ and $O(h)$ for Darcy-related quantities, with no pollution from the Darcy approximation to the Stokes velocity, through a carefully constructed Stokes–Darcy Ritz projection and rigorous error analysis. The results hold for both dynamic and steady Stokes–Darcy models and are supported by numerical experiments that confirm the theory and demonstrate the method’s efficiency. This provides a reliable, efficient framework for non-uniform, fully-mixed FEMs in multiphysics Stokes–Darcy simulations of practical relevance.

Abstract

This paper is concerned with non-uniform fully-mixed FEMs for dynamic coupled Stokes-Darcy model with the well-known Beavers-Joseph-Saffman (BJS) interface condition. In particular, a decoupled algorithm with the lowest-order mixed non-uniform FE approximations (MINI for the Stokes equation and RT0-DG0 for the Darcy equation) and the classical Nitsche-type penalty is studied. The method with the combined approximation of different orders is commonly used in practical simulations. However, the optimal error analysis of methods with non-uniform approximations for the coupled Stokes-Darcy flow model has remained challenging, although the analysis for uniform approximations has been well done. The key question is how the lower-order approximation to the Darcy flow influences the accuracy of the Stokes solution through the interface condition. In this paper, we prove that the decoupled algorithm provides a truly optimal convergence rate in L^2-norm in spatial direction: O(h^2) for Stokes velocity and O(h) for Darcy flow in the coupled Stokes-Darcy model. This implies that the lower-order approximation to the Darcy flow does not pollute the accuracy of numerical velocity for Stokes flow. The analysis presented in this paper is based on a well-designed Stokes-Darcy Ritz projection and given for a dynamic coupled model. The optimal error estimate holds for more general combined approximations and more general coupled models, including the corresponding model of steady-state Stokes-Darcy flows and the model of coupled dynamic Stokes and steady-state Darcy flows. Numerical results confirm our theoretical analysis and show that the decoupled algorithm is efficient.

Optimal analysis of penalized lowest-order mixed FEMs for the Stokes-Darcy model

TL;DR

This work addresses the dynamic Stokes–Darcy coupling with Beavers–Joseph–Saffman interface conditions by introducing a decoupled, fully-mixed FEM that uses non-uniform lowest-order approximations (MINI for Stokes and RT0–DG0 for Darcy) and a Nitsche-type penalty for the interface. The authors prove true optimal spatial convergence rates for the coupled system, showing accuracy for the Stokes velocity in and for Darcy-related quantities, with no pollution from the Darcy approximation to the Stokes velocity, through a carefully constructed Stokes–Darcy Ritz projection and rigorous error analysis. The results hold for both dynamic and steady Stokes–Darcy models and are supported by numerical experiments that confirm the theory and demonstrate the method’s efficiency. This provides a reliable, efficient framework for non-uniform, fully-mixed FEMs in multiphysics Stokes–Darcy simulations of practical relevance.

Abstract

This paper is concerned with non-uniform fully-mixed FEMs for dynamic coupled Stokes-Darcy model with the well-known Beavers-Joseph-Saffman (BJS) interface condition. In particular, a decoupled algorithm with the lowest-order mixed non-uniform FE approximations (MINI for the Stokes equation and RT0-DG0 for the Darcy equation) and the classical Nitsche-type penalty is studied. The method with the combined approximation of different orders is commonly used in practical simulations. However, the optimal error analysis of methods with non-uniform approximations for the coupled Stokes-Darcy flow model has remained challenging, although the analysis for uniform approximations has been well done. The key question is how the lower-order approximation to the Darcy flow influences the accuracy of the Stokes solution through the interface condition. In this paper, we prove that the decoupled algorithm provides a truly optimal convergence rate in L^2-norm in spatial direction: O(h^2) for Stokes velocity and O(h) for Darcy flow in the coupled Stokes-Darcy model. This implies that the lower-order approximation to the Darcy flow does not pollute the accuracy of numerical velocity for Stokes flow. The analysis presented in this paper is based on a well-designed Stokes-Darcy Ritz projection and given for a dynamic coupled model. The optimal error estimate holds for more general combined approximations and more general coupled models, including the corresponding model of steady-state Stokes-Darcy flows and the model of coupled dynamic Stokes and steady-state Darcy flows. Numerical results confirm our theoretical analysis and show that the decoupled algorithm is efficient.

Paper Structure

This paper contains 9 sections, 5 theorems, 99 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Algorithm and main results
  3. Notation and weak formulation
  4. Fully discrete algorithm with the non-uniform approximations
  5. Main results
  6. Analysis
  7. Proof of Theorem \ref{['3-1']}
  8. Numerical results
  9. Declarations

Key Result

Theorem 2.1

Suppose that the system e-f--initial has a unique solution $({\boldsymbol{u}}_{f}, {\boldsymbol{u}}_{p}, p_{f}, \phi_{p})$ satisfying the regularity regularity. Then the fully discrete system decoupledF--decoupledP yields a unique solution $({\boldsymbol{u}}_{fh}^{m}, {\boldsymbol{u}}_{ph}^{m}, p^{m where $C_0$ is a positive constant independent of $h$ and $\tau$.

Figures (4)

  • Figure 1: A sketch of the free flow region $\Omega_f$, porous media flow region $\Omega_p$ and the interface $\Gamma$.
  • Figure 2: Left: the geometric location of the simulation domain; Right: quasi-uniform grid.
  • Figure 3: The flow speed, streamlines and pressure around horizontal open-hole attached with vertical wellbore completion for reservoir velocity $\Theta_1=1$. Left: flow speed streamlines; Right: pressure.
  • Figure 4: The flow speed and streamlines around horizontal open-hole attached with vertical production wellbore completion for different reservoir velocities. Left: $\Theta_2=3$; Right: $\Theta_3=5$.

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1: acosta2011errorlayton2002coupling
  • Lemma 3.2
  • Lemma 3.3: sun2022new