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:我们考虑反复解决由固定式斯托克斯-达西方程式产生的两个层面的庞大和稀少的双马座点系统的问题,这两个层面由马克-卡尔(MAC)的有限差异法分解,我们分析了少数理想的区块先决条件者的精华价值分布,然后得出基于Schur补充物近似值的实用先决条件,这种近似值产生于双马座点矩阵的块状分解。我们表明,将接口条件纳入先决条件是追求可扩缩性的关键。数字结果显示,我们具有先决条件的GMRES解决方案解决者有着良好的趋同行为,并表明拟议的先决条件对于问题的物理参数具有很强的说服力。</s>