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.

翻译：自由流动与多孔介质耦合系统广泛存在于各类工程与环境应用中。对于层流状态，此类系统在自由流动区域由Stokes方程描述，在多孔介质区域由达西定律描述。在流体-多孔介质界面需要采用适当的耦合条件集。Stokes-Darcy问题的离散化会产生大规模、稀疏、病态的线性系统，且根据界面条件的不同可能呈现非对称性。因此需要稳健高效的预处理技术来加速所采用的Krylov方法的收敛。本研究针对耦合Stokes-Darcy问题，采用二阶MAC格式，开发并研究了块对角、块三角及约束预处理方法。我们应用两种经典的耦合条件集，分别考虑切向速度的Beavers-Joseph条件和Beavers-Joseph-Saffman条件。对于Beavers-Joseph界面条件，所得系统为非对称形式，因此两种情形均采用GMRES方法。通过对精确版本预处理子进行谱分析，确定了特征值聚集区域与边界。此外，面向实际应用我们开发了高效的非精确版本预处理子。数值实验证明了所提预处理方法的有效性与鲁棒性。