Block Preconditioners for the Marker-and-Cell Discretization of the Stokes–Darcy Equations

We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes–Darcy equations in two dimensions, discretized by the marker-and-cell finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results show good convergence behavior of our preconditioned GMRES solver and demonstrate robustness of the proposed preconditioner with respect to the physical parameters of the problem.