We consider the coupled Stokes-Darcy system in a specified domain with two subregions. This is a system of partial differential equations describing two flows. In one subregion of the domain, freely flowing fluid is governed by the Stokes equations. In the other subregion there is flow governed by Darcy's law. The two flows are then coupled together by conditions on the interface of the two flows. This model can physically represent fluid percolating through a porous medium.
The problem is solved numerically using the finite element method. The Stokes domain is discretized using standard, continuous finite element spaces that satisfy and inf-sup condition. In the Darcy domain we consider both continuous functions and discontinuous polynomials (discontinuous Galerkin methods). In both cases the discretization leads to a system of equations that is large, sparse, non-symmetric (and non-symmetrizable in the DG case) and of saddle point form.
We propose solving the system of equations using preconditioned GMRES with an indefinite (or constraint) preconditioner which mimics the structure of the original system matrix. We prove that the convergence of GMRES using this preconditioner is bounded independently of the underlying mesh discretization. We present numerical results showing that the indefinite preconditioner outperforms both standard block diagonal and block triangular preconditioners both with respect to iteration count and CPU times.