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Analysis and Elimination of Numerical Pressure Dependency in Coupled Stokes-Darcy Problem

Jiachuan Zhang

TL;DR

The paper addresses the sensitivity of velocity accuracy to pressure in mixed FEM for the Stokes-Darcy system. It introduces an auxiliary velocity projection $S_h\boldsymbol{u}$ to decouple velocity error from pressure and develops a divergence-free reconstruction operator $\Pi_h$ to enforce exact discrete divergence and interface continuity, yielding pressure-robust discretizations. Theoretical a priori estimates show that the pressure-dependent consistency errors vanish with $\Pi_h$, and numerical tests confirm dramatic reductions in velocity error under high pressure or low viscosity, while maintaining competitive computational costs. The work broadens pressure-robust strategies to coupled multi-physics settings with higher-order and 3D potential, offering a robust framework for simulations in porous media and free-fluid coupling. It paves the way for extensions to time-dependent problems and nonlinear interactions, enhancing reliability and efficiency of multi-physics simulations.

Abstract

This paper presents a pressure-robust mixed finite element method (FEM) for the coupled Stokes-Darcy system. We revisits the rigorous theoretical framework of Layton et al. [2002], where velocity and pressure errors are coupled, masking pressure's influence on velocity accuracy. To investigate the pressure dependency, we introduce a auxiliary velocity projection that preserves discrete divergence and interface continuity constraints. By analyzing the difference between the discrete and projected velocities, we rigorously prove that classical FEM incurs pressure-dependent consistency errors due to inexact divergence enforcement and approximate interface conditions. To eliminate these errors, we design a pressure-robust method using divergence-free reconstruction operator, which enforce exact divergence constraint and interface continuity. Numerical examples confirm the theory: under high-pressure or low-viscosity, the proposed method reduces velocity errors by orders of magnitude compared to classical method.

Analysis and Elimination of Numerical Pressure Dependency in Coupled Stokes-Darcy Problem

TL;DR

The paper addresses the sensitivity of velocity accuracy to pressure in mixed FEM for the Stokes-Darcy system. It introduces an auxiliary velocity projection to decouple velocity error from pressure and develops a divergence-free reconstruction operator to enforce exact discrete divergence and interface continuity, yielding pressure-robust discretizations. Theoretical a priori estimates show that the pressure-dependent consistency errors vanish with , and numerical tests confirm dramatic reductions in velocity error under high pressure or low viscosity, while maintaining competitive computational costs. The work broadens pressure-robust strategies to coupled multi-physics settings with higher-order and 3D potential, offering a robust framework for simulations in porous media and free-fluid coupling. It paves the way for extensions to time-dependent problems and nonlinear interactions, enhancing reliability and efficiency of multi-physics simulations.

Abstract

This paper presents a pressure-robust mixed finite element method (FEM) for the coupled Stokes-Darcy system. We revisits the rigorous theoretical framework of Layton et al. [2002], where velocity and pressure errors are coupled, masking pressure's influence on velocity accuracy. To investigate the pressure dependency, we introduce a auxiliary velocity projection that preserves discrete divergence and interface continuity constraints. By analyzing the difference between the discrete and projected velocities, we rigorously prove that classical FEM incurs pressure-dependent consistency errors due to inexact divergence enforcement and approximate interface conditions. To eliminate these errors, we design a pressure-robust method using divergence-free reconstruction operator, which enforce exact divergence constraint and interface continuity. Numerical examples confirm the theory: under high-pressure or low-viscosity, the proposed method reduces velocity errors by orders of magnitude compared to classical method.
Paper Structure (9 sections, 6 theorems, 90 equations, 7 figures, 2 tables)

This paper contains 9 sections, 6 theorems, 90 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem statements and variational formulation
  3. Classical FEM discretization
  4. Pressure-robust discretization
  5. Numerical experiments
  6. Basis functions, degree of freedom, and computing $(\boldsymbol{f},\Pi_h\boldsymbol{\psi}_h)$
  7. Example for different pressure
  8. Example for different viscosity
  9. Conclusion

Key Result

Lemma 3.1

There exists an operator $\Upsilon_h^s: V^s\rightarrow V_1^s\subset V_h^s$ satisfying, for any $T\in \mathcal{T}_h(\Omega^s)$ and all $\boldsymbol{v}^s\in V^s$, where $V_1^s=\{\boldsymbol{v}_h^s\in V^s~|~\boldsymbol{v}_{h|T}^s\in \boldsymbol{P}_1^+(T), T\in\mathcal{T}_h(\Omega^s)\}$ with $\boldsymbol{P}_1^+(T)=[P_1(T)]^N\oplus span\{\boldsymbol{p}_1,\cdots,\boldsymbol{p}_{N+1}\}$.

Figures (7)

  • Figure 5.1: Convergence rates of velocity for classical method (top) and pressure-robust method (bottom) with $\gamma=1$ (left) and $10^5$ (right), respectively, in Example \ref{['ex1']}.
  • Figure 5.2: Convergence rates of pressure for classical method (top) and pressure-robust method (bottom) with $\gamma=1$ (left) and $10^5$ (right), respectively, in Example \ref{['ex1']}.
  • Figure 5.3: Total errors (top) and component errors (bottom) for classical method (left) and pressure-robust method (right) with $d.o.f=17763$ in Example \ref{['ex1']}.
  • Figure 5.4: Streamlines of velocity for classical method (left), exact solution (middle), and pressure-robust method (right) with $\mu=10^{-6}$ and $d.o.f=8931$, in Example \ref{['ex2']}.
  • Figure 5.5: Convergence rates of velocity for classical method (top) and pressure-robust method (bottom) with $\mu=10^{-6}$ (left), $1$ (middle), and $10^6$ (right), respectively, in Example \ref{['ex2']}.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 5 more