In this article a domain decomposition method is proposed to solve a time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition. Robin boundary conditions between the Navier-Stokes domain and Darcy domain are constructed by directly re-organizing the terms in the three interface conditions, including the Beavers-Joseph condition. In order to avoid the traditional iteration for the domain decomposition method at each time step, the interface information, which is needed for the Robin type transmission conditions at the current time step, is directly predicted based on the numerical solution of the previous time steps. Backward Euler scheme is first utilized for the temporal discretization while finite elements are used for the spatial discretization. The convergences of this domain decomposition method are rigorously analyzed for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition. The major difficulties in the analysis arise from nonlinear terms and Beavers-Joseph interface condition, including a series of technical treatments and the final special norm used in the discrete Gronwall's inequality for the analysis of full discretization. Based on the above preparation, we further develop a Lagrange multiplier method under the framework of the domain decomposition method to overcome the difficulty of non-unique solutions arising from the defective boundary condition. One interesting finding of this paper is that the Lagrange multipliers are time dependent functions instead of constants. In order to improve the accuracy order for the temporal discretization, a three-step backward differentiation scheme is used to replace the backward Euler scheme. Compared with the first scheme, the second one allows us to use the relative larger time step to reduce the computational cost while keeping the same accuracy. Numerical examples are provided to illustrate the features of the proposed method.

本文提出了一种域分解方法来求解具有Beavers-Joseph界面条件和缺陷边界条件的时变Navier-Stokes-Darcy模型。Navier-Stokes域和Darcy域之间的Robin边界条件是通过直接重新组织三个界面条件（包括Beavers-Joseph条件）中的项来构造的。为了避免在每个时间步进行域分解方法的传统迭代，根据先前时间步的数值解直接预测当前时间步的Robin类型传输条件所需的接口信息。首先将后向Euler方案用于时间离散化，而将有限元用于空间离散化。对于Beavers-Joseph界面条件下的时间相关Navier-Stokes-Darcy模型，严格分析了该域分解方法的收敛性。分析中的主要困难来自非线性项和Beavers-Joseph界面条件，包括一系列技术处理以及离散Gronwall不等式中用于完全离散化分析的最终特殊规范。在上述准备的基础上，我们进一步在域分解方法的框架下开发了拉格朗日乘数法，以克服边界条件有缺陷产生非唯一解的困难。本文的一个有趣发现是，拉格朗日乘数是时间相关函数，而不是常量。为了提高时间离散化的准确性，使用三步向后差分方案代替反向欧拉方案。与第一种方案相比，第二种方案允许我们使用相对较大的时间步长来降低计算成本，同时保持相同的精度。数值例子说明了该方法的特点。