Parameter-robust Preconditioners for the Stokes-Darcy Coupled Problem without Fractional Operators

TL;DR

This paper develops a parameter-robust preconditioner for the Stokes-Darcy coupled problem that avoids fractional operators, ensuring stability and efficiency across various boundary conditions and discretizations.

Contribution

It introduces a unified, fractional-operator-free preconditioning approach for the Stokes-Darcy system, validated through theoretical analysis and numerical experiments.

Findings

Preconditioner is robust to physical parameters and mesh size.

Numerical results confirm theoretical robustness.

Method applies to both conforming and nonconforming finite-element methods.

Abstract

We consider the Stokes-Darcy coupled problem, which models the interaction between free-flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element spaces, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both $H(\operatorname{div})$-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioners with respect to the physical parameters and mesh size.