Robust and Efficient Preconditioners for Stokes–Darcy Problems

Coupled systems of porous media and free flow can be modelled by the Stokes equations in the free-flow domain, Darcy’s law in the porous medium, and an appropriate set of coupling conditions on the fluid–porous interface. Discretisation of the coupled Stokes–Darcy problem leads to a large, sparse, ill-conditioned, and nonsymmetric linear system. We discretise the system using the MAC scheme, i.e., the finite volume method on staggered grids. To accelerate convergence of the GMRES method, efficient preconditioners are needed. We propose a block diagonal, a block triangular and a constraint preconditioner for the Stokes–Darcy problem with the classical set of coupling conditions based on the Beavers–Joseph condition and the generalised coupling conditions which were developed for arbitrary flow to the interface. We show the robustness and efficiency of the proposed preconditioners in numerical experiments.