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

This paper analyses the classical mixed finite element method (FEM) and a pressure-robust variant with divergence-free reconstruction operators for the coupled Stokes-Darcy problem. Its main contribution is to provide viscosity-explicit a priori error estimates that clearly distinguish the pressure dependence of the two discretizations: the velocity error of the classical scheme depends on both the exact pressure and the viscosity, whereas the pressure-robust method eliminates both entirely. Moreover, we derive pressure error estimates and quantify their dependence on the exact solution and model parameters. Two-dimensional numerical experiments validate the theoretical findings, including higher-order tests up to polynomial degree three and a lid-driven cavity benchmark with a piecewise linear interface. The implementation code is made publicly available to facilitate reproducibility.

💡 Research Summary

The paper addresses the numerical treatment of the coupled Stokes–Darcy system, a model that describes the interaction between a free‑flow (Stokes) region and a porous‑media (Darcy) region across an interface Γ. While mixed finite‑element methods (FEM) are widely used for this problem, standard formulations suffer from a pressure‑dependent velocity error: the velocity error scales with the inverse of the fluid viscosity μ and with the approximation error of the pressure. This dependence becomes critical in low‑viscosity regimes, leading to loss of accuracy and possible instability.

The authors first present a rigorous viscosity‑explicit a‑priori error analysis for the classical mixed FEM. Using conforming shape‑regular triangulations, they employ continuous polynomial spaces enriched with bubble functions for the Stokes velocity (degree k ≥ 2) and Raviart‑Thomas spaces for the Darcy velocity, together with matching pressure spaces. The discrete variational problem (3.2) satisfies the LBB condition, but the consistency error H contains a term that depends on the exact pressure. By introducing a pressure‑related functional ϑ_p(·) they show that the velocity error bound contains a term μ⁻¹‖p‑p_h‖_0, confirming the classical scheme’s pressure‑sensitivity.

To eliminate this undesirable dependence, the paper adopts the pressure‑robust paradigm based on divergence‑free reconstruction operators. Building on the ideas of Linke and collaborators, the authors define a reconstruction operator Υ_{sh} for the Stokes velocity space and, more importantly, a global auxiliary projector S_h : V(g) → V_h(g) that satisfies a(S_h u, ψ_h) + b(ψ_h, δ_h) = a(u, ψ_h) for all test functions ψ_h and enforces the discrete divergence constraint. The key property of S_h is that it modifies only the right‑hand side of the variational formulation, leaving the left‑hand side unchanged, thereby preserving the original physics while making the test functions exactly divergence‑free.

With this reconstruction, the authors derive a new error estimate for the pressure‑robust scheme: ‖u – u_h‖X ≤ C inf{v_h∈V_h}‖u – v_h‖_X, which is independent of both μ and the exact pressure. The velocity error now depends solely on the approximation capabilities of the chosen finite‑element spaces. A complementary pressure error bound is also obtained: ‖p – p_h‖_0 ≤ C ( μ‖u – S_h u‖_X + approximation terms ), showing that pressure accuracy is controlled by the velocity reconstruction error but does not feed back into the velocity error.

The theoretical findings are substantiated by extensive two‑dimensional numerical experiments. First, convergence tests for polynomial degrees k = 1, 2, 3 confirm the predicted rates for both schemes. However, when μ is reduced to 10⁻⁴, the classical method’s velocity error grows dramatically, whereas the pressure‑robust method maintains the same error level, illustrating its robustness. Second, a lid‑driven cavity benchmark with a piecewise‑linear interface demonstrates that the pressure‑robust method reproduces the expected vortex structure without spurious pressure‑induced artifacts, while the classical method shows noticeable distortion near the interface. All simulations are performed with an open‑source implementation based on the FEniCS platform; the code is publicly released to promote reproducibility.

In conclusion, the paper makes three major contributions: (1) it provides the first viscosity‑explicit error analysis that clearly separates pressure and viscosity effects for the coupled Stokes–Darcy problem; (2) it extends the divergence‑free reconstruction technique to a mixed interface problem, achieving full pressure‑robustness for the velocity field; (3) it validates the theory with high‑order numerical tests and a realistic benchmark, and supplies reproducible software. The authors suggest future work on three‑dimensional extensions, time‑dependent problems, and nonlinear porous‑media models such as the Forchheimer law.