Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Highlights: • A monolithic multigrid method is developed for the coupled multi-physics system. • Unknowns from different subsystems are defined at the same point at the interface. • Textbook multigrid efficiency is obtained for this highly nontrivial coupled system. • The asymptotic convergence of the coupled system can be predicted well by LFA. • The multigrid method performs highly satisfactory for realistically small parameters. The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients.