Inexact block triangular preconditioners for double saddle-point systems arising from coupled Stokes–Darcy model

The interaction between fluids in porous media and free flow regions is significant in various fields such as geology, environmental engineering, and petroleum engineering. This problem can be modeled using a coupled system of Stokes equations and Darcy’s law. Finite element discretization of the coupled Stokes–Darcy model results in a double saddle-point system. In this work, we propose a class of inexact block three-by-three triangular preconditioners for these discretized saddle-point systems by fully utilizing their special structure. When used to precondition Krylov subspace methods, each step requires solving only three simple symmetric positive definite linear subsystems. Additionally, we derive explicit and sharp spectral bounds for the preconditioned matrices and show that all eigenvalues have positive real parts. Numerical experiments demonstrate the effectiveness and robustness of our preconditioners in solving the coupled Stokes–Darcy model.