A parameter-robust iterative method for Stokes–Darcy problems retaining local mass conservation

¹ University of Stuttgart, Department of Hydromechanics and Modelling of Hydrosystems, Pfaffenwaldring 61, Stuttgart 70569, Germany
² KTH Royal Institute of Technology, Department of Mathematics, Lindstedtsvägen 25, Stockholm 114 28, Sweden

^* Corresponding author: wietse@kth.se

Received: 22 July 2019
Accepted: 6 May 2020

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.