An efficient coupled block preconditioner for the Marker-and-Cell discretization of the Stokes-Darcy equations

We consider the iterative solution of the large, sparse and nonsymmetric double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and- Cell (MAC) finite difference method. Existing block triangular preconditioners rest on approximations of two Schur complements, the second of which is hard to approximate uniformly in the physical parameters. We propose instead a coupled preconditioner that keeps the interface coupling and regularizes the constraint block by a small multiple of the identity, so that the second Schur complement never has to be approximated. We analyze the spectrum of the preconditioned matrix, both in the ideal case and for the inexact version used in computations, and derive from it a rule for scaling the regularization parameter with the physical parameters. Numerical results show mesh-independent convergence of preconditioned GMRES and robustness with respect to the viscosity and the permeability.