FETI and BDD preconditioners for Stokes-Mortar-Darcy systems. (English) Zbl 1189.35226

Galvis, Juan; Sarkis, Marcus

Commun. Appl. Math. Comput. Sci. 5, 1-30 (2010).

Summary: 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 \(C_1(1 + (1/\kappa))\) for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order \(C_2((\kappa + 1)/(\kappa + (h^p)^2))\) 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.

MSC:

35Q30	Navier-Stokes equations
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
76D07	Stokes and related (Oseen, etc.) flows

Keywords:

Stokes-Darcy coupling; mortar; balancing domain decomposition; FETI; saddle point problems; nonmatching grids; discontinuous coefficients; mortar elements