Block preconditioners for the Marker and Cell discretization of the Stokes-Darcy equations

We develop block preconditioners for solving the Stokes-Darcy equations, discretized by the Marker and Cell (MAC) finite difference method. The discretization leads to a mildly nonsymmetric double saddle-point linear system. We identify numerical properties and exploit the sparsity structure of the matrix, for the purpose of developing a fast preconditioned iterative solution procedure. The proposed preconditioners are based on approximations of two Schur complements that arise in decompositional relations associated with the double saddle-point matrix and its blocks. We analyze the eigenvalue distribution of the preconditioned matrices with respect to the physical parameters of the problem, and show that the eigenvalues are strongly clustered. Consequently, preconditioned GMRES appears to be relatively insensitive to the mesh size and the physical parameters involved. Numerical results validate our theoretical observations.