A Domain Decomposition Method for the Steady-State Navier--Stokes--Darcy Model with Beavers--Joseph Interface Condition

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier--Stokes--Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers--Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers--Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1--25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239--4256]. In this paper, the well-posedness of the Navier--Stokes--Darcy model with Beavers--Joseph condition is analyzed by using a branch of nonsingular solutions. By following the idea in [Y. Cao et al., Numer. Math., 117 (2011), pp. 601--629], the three physical interface conditions are utilized together to construct the Robin-type boundary conditions on the interface and decouple the two physics which are described by Navier--Stokes and Darcy equations, respectively. Then the corresponding multiphysics DDM is proposed and analyzed. Three numerical experiments using finite elements are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis.