The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss–Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, local Fourier analysis is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity, which are relevant for applications.

Additional Metadata
Keywords Darcy equation, porous medium, Stokes equation, free flow, coupling, interface conditions, multigrid method, Uzawa smoother, local Fourier analysis
MSC Multigrid methods; domain decomposition (msc 65N55), Iterative methods for linear systems (msc 65F10)
Persistent URL dx.doi.org/10.1137/16M1076514
Journal SIAM Journal on Scientific Computing
Grant This work was funded by the European Commission 7th Framework Programme; grant id h2020/705402 - Efficient numerical methods for deformable porous media. Application to carbon dioxide storage. (poro sos)
Citation
Luo, P, Rodrigo, C, Gaspar, F.J, & Oosterlee, C.W. (2017). Uzawa smoother in multigrid for the coupleD porous medium and stokes flow system. SIAM Journal on Scientific Computing, 39(5), S633–S661. doi:10.1137/16M1076514