Preprint A563/2007
FETI and BDD Preconditioners for Stokes-Mortar-Darcy Systems
Marcus Sarkis | Galvis, Juan
Keywords: Stokes Darcy coupling | mortar | BDD | FETI | saddle point problems | nonmatching grids | discontinuous coefficients | mortar elements
We consider the coupling across an interface of a fluid flow and a porous media flow. The differential equations involve Stokes equations in the fluid region and Darcy equations in the porous region, and coupled through an interface with Beaver-Joseph-Saffman transmission conditions. The discretization consists of $P2/P1$ triangular Taylor-Hood finite elements in the fluid region, the lowest order triangular Raviart-Thomas finite elements in the porous region, and the mortar piecewise constant Lagrange multipliers on the interface and we allow nonmatching meshes across the interface. Due to the small values of the permeability parameter $\kappa$ of the porous medium, the resulting discrete symmetric saddle point system is very ill conditioned. We design and analyze a preconditioners based on the Finite Element by Tearing and Interconnecting (FETI) and Balancing Domain Decomposition (BDD) preconditioners and derive a condition number estimate of order $C_1(1+\frac{1}{\kappa})$. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order $C_2(\frac{\kappa+1}{\kappa+(h^\pr)^2} )$ for the FETI preconditioner. Here $h^\pr$ is the mesh size of the porous side triangulation. The constants $C_1$ and $C_2$ are independent of the permeability $\kappa$, the fluid viscosity $\nu$, and the mesh ratio across the interface. Numerical experiments confirm the sharpness of the theoretical estimates.