We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results show good convergence behavior of our preconditioned GMRES solver and demonstrate robustness of the proposed preconditioner with respect to the physical parameters of the problem.

翻译：我们考虑求解由二维定常斯托克斯-达西方程离散化产生的大规模稀疏双鞍点系统的问题，该系统采用标记-网格（MAC）有限差分方法进行离散。分析了几种理想块预条件子的特征值分布。随后，基于双鞍点矩阵块分解中出现的舒尔补近似，推导了实用型预条件子。研究表明，将界面条件纳入预条件子是实现可扩展性的关键。数值结果表明，我们所预处理的GMRES求解器具有良好的收敛行为，并且所提出的预条件子对问题的物理参数表现出鲁棒性。