The coupled Darcy–Stokes problem is widely used for modeling fluid transport in physical systems consisting of a porous part and a free part. In this work we consider preconditioners for monolithic solution algorithms of the coupled Darcy–Stokes problem, where the Darcy problem is in primal form. We employ the operator preconditioning framework and utilize a fractional solver at the interface between the problems to obtain order optimal schemes that are robust with respect to the material parameters, i.e. the permeability, viscosity and Beavers–Joseph–Saffman condition. Our approach is similar to that of Holter et al. (2020), but since the Darcy problem is in primal form, expressing mass conservation at the interface involves the normal derivative, which introduces some mathematical challenges. These challenges will be specifically addressed in this paper, in particular we will employ fractional Laplacians at the interface. Numerical experiments illustrating the performance are provided. The preconditioner is posed in non-standard Sobolev spaces which may be perceived as an obstacle for its use in applications. However, we detail the implementational aspects and show that the preconditioner is quite feasible to realize in practice.

达西－斯托克斯耦合问题已广泛用于对由多孔部分和自由部分组成的物理系统中的流体传输进行建模。在这项工作中，我们考虑耦合Darcy–Stokes问题的整体解决方案算法的预处理器，其中Darcy问题为原始形式。我们采用操作员预处理框架，并在问题之间的界面处使用分数求解器，以获得关于材料参数（例如渗透率，粘度和Beavers–Joseph–Saffman条件）稳健的有序优化方案。我们的方法类似于Holter等人的方法。（2020年），但由于达西问题是原始形式，在界面上表示质量守恒涉及正态导数，这引入了一些数学挑战。这些挑战将在本文中专门解决，特别是在接口处将使用分数拉普拉斯算子。提供了说明性能的数值实验。预处理器放置在非标准的Sobolev空间中，这可能会被视为其在应用中使用的障碍。但是，我们详细介绍了实现方面，并显示了预处理器在实践中完全可行。