The friction-type (or called barrier-type) leak interface condition (FLIC) is proposed to model the viscous fluid through a perforated membrane with a threshold permeability, where the flow passes through the perforations only when the stress difference on the membrane is above a threshold. This work establishes a comprehensive study on several numerical approaches for the Stokes/Stokes coupled flow under the FLIC, including the projection, regularization (or called penalty), and domain decomposition methods. In the continuous sense, first, we revisit the well-posedness, introduce the Lagrange multiplier formulation, and show the convergence of the projection method, i.e., the Uzawa algorithm. Second, we approximate the variational inequality by the regularization technique. We derive the approximation error and discuss the applicability of the Picard iteration to the nonlinear regularization problem. And third, a domain decomposition algorithm is proposed to decouple the system into two Stokes problems with the Dirichlet and friction-type leak boundary conditions on \(\Gamma \), respectively. In the discrete sense, we apply the finite element method with P1b/P1-element and enforce the interface condition by mass-lumping technique. We obtain the error estimates of the finite element approximation. The convergence of the discrete version of the projection methods (including the Uzawa and Active/Inactive set algorithms), the regularization problem, and the domain decomposition algorithm are investigated as with the continuous cases. We carry out several numerical experiments to confirm the theoretical results.

摩擦型（或称为屏障型）泄漏界面提出条件 (FLIC) 来模拟通过具有阈值渗透率的穿孔膜的粘性流体，其中仅当膜上的应力差高于阈值时，流体才通过穿孔。这项工作对 FLIC 下 Stokes/Stokes 耦合流的几种数值方法进行了全面研究，包括投影、正则化（或称为惩罚）和域分解方法。在连续意义上，首先，我们重新审视适定性，引入拉格朗日乘数公式，并展示投影方法即Uzawa 算法的收敛性。其次，我们通过正则化技术来近似变分不等式。我们推导了近似误差并讨论了 Picard 迭代对非线性正则化问题的适用性。\(\Gamma \)，分别。在离散意义上，我们应用具有 P1b/P1 单元的有限元方法，并通过质量集中技术加强界面条件。我们获得了有限元近似的误差估计。投影方法（包括 Uzawa 和活动/非活动集算法）的离散版本的收敛性、正则化问题和域分解算法与连续情况一样进行了研究。我们进行了几个数值实验来证实理论结果。