A Parameter-Robust Iterative Method for Stokes-Darcy Problems Retaining Local Mass Conservation

TL;DR

This paper introduces a robust iterative method for coupled Stokes-Darcy problems that ensures local mass conservation and maintains performance across various discretization schemes, supported by theoretical analysis and numerical validation.

Contribution

A novel iterative scheme for Stokes-Darcy coupling that guarantees local mass conservation and robustness, applicable to diverse discretization methods.

Findings

The scheme is well-posed in weighted norms.

Performance is robust against material and discretization parameters.

Numerical experiments confirm theoretical results.

Abstract

We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analysis, the performance of the iterative scheme is shown to be robust with respect to material and discretization parameters. By construction, the scheme is applicable to a wide range of locally conservative discretization schemes and we consider explicit examples in the framework of Mixed Finite Element methods. Finally, the theoretical results are confirmed with the use of numerical experiments.