A coupling concept for Stokes-Darcy systems: The ICDD method

We present a coupling framework for Stokes-Darcy systems valid for arbitrary flow direction at low Reynolds numbers and for isotropic porous media. The proposed method is based on an overlapping domain decomposition concept to represent the transition region between the free-fluid and the porous-medium regimes. Matching conditions at the interfaces of the decomposition impose the continuity of velocity (on one interface) and pressure (on the other one) and the resulting algorithm can be easily implemented in a non-intrusive way. The numerical approximations of the fluid velocity and pressure obtained by the studied method converge to the corresponding counterparts computed by direct numerical simulation at the microscale, with convergence rates equal to suitable powers of the scale separation parameter ε in agreement with classical results in homogenization

The Interface Control Domain Decomposition method for Stokes-Darcy coupling

The interface control domain decomposition (ICDD) method of Discacciati, Gervasio, and Quarteroni [SIAM J. Control Optim., 51 (2013), pp. 3434--3458, doi:10.1137/120890764; J. Coupled Syst. Multiscale Dyn., 1 (2013), pp. 372--392, doi:10.1166/jcsmd.2013.1026] is proposed here to solve the coupling between Stokes and Darcy equations. According to this approach, the problem is formulated as an optimal control problem whose control variables are the traces of the velocity and the pressure on the internal boundaries of the subdomains that provide an overlapping decomposition of the original computational domain. A theoretical analysis is carried out, and the well-posedness of the problem is proved under certain assumptions on both the geometry and the model parameters. An efficient solution algorithm is proposed, and several numerical tests are implemented. Our results show the accuracy of the ICDD method, its computational efficiency, and robustness with respect to the different parameters involved (grid size, polynomial degrees, permeability of the porous domain, thickness of the overlapping region). The ICDD approach turns out to be more versatile and easier to implement than the celebrated model based on the Beavers, Joseph, and Saffman coupling conditions