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.