Efficient preconditioners for coupled Stokes-Darcy problems with MAC scheme: Spectral analysis and numerical study

Abstract

Coupled systems of free flow and porous media arise in a variety of technical and environmental applications. For laminar flow regimes, such systems are described by the Stokes equations in the free-flow region and Darcy's law in the porous medium. An appropriate set of coupling conditions is needed on the fluid-porous interface. Discretisations of the Stokes-Darcy problems yield large, sparse, ill-conditioned, and, depending on the interface conditions, non-symmetric linear systems. Therefore, robust and efficient preconditioners are needed to accelerate convergence of the applied Krylov method. In this work, we consider the second order MAC scheme for the coupled Stokes-Darcy problems and develop and investigate block diagonal, block triangular and constraint preconditioners. We apply two classical sets of coupling conditions considering the Beavers-Joseph and the Beavers-Joseph-Saffman condition for the tangential velocity. For the Beavers-Joseph interface condition, the resulting system is non-symmetric, therefore GMRES method is used for both cases. Spectral analysis is conducted for the exact versions of the preconditioners identifying clusters and bounds. Furthermore, for practical use we develop efficient inexact versions of the preconditioners. We demonstrate effectiveness and robustness of the proposed preconditioners in numerical experiments.