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. Numer-ical 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.