A domain decomposition method for the time-dependent Navier-Stokes-Darcy model with Beavers-Joseph interface condition and defective boundary condition

Abstract

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.