FETI and BDD preconditioners for Stokes–Mortar–Darcy Systems

Abstract

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, Darcy equations in the porous region, plus a coupling 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. We allow for 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 preconditioners based on the finite element by tearing and interconnecting (FETI) and balancing domain decomposition (BDD) methods and derive a condition number estimate of order $C_1\left(1+\left(1∕\kappa \right)\right)$ for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order $C_2\left(\left(\kappa +1\right)∕\left(\kappa +{\left(h^p\right)}^2\right)\right)$ for the FETI preconditioner. Here $h^p$ 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.