Vol. 5, No. 1, 2010

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
FETI and BDD preconditioners for Stokes–Mortar–Darcy Systems

Juan Galvis and Marcus Sarkis

Vol. 5 (2010), No. 1, 1–30
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 κ 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 C1(1 + (1κ)) for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order C2((κ + 1)(κ + (hp)2)) for the FETI preconditioner. Here hp is the mesh size of the porous side triangulation. The constants C1 and C2 are independent of the permeability κ, the fluid viscosity ν, and the mesh ratio across the interface. Numerical experiments confirm the sharpness of the theoretical estimates.

Keywords
Stokes–Darcy coupling, mortar, balancing domain decomposition, FETI, saddle point problems, nonmatching grids, discontinuous coefficients, mortar elements
Mathematical Subject Classification 2000
Primary: 35Q30, 65N22, 65N30, 65N55, 76D07
Milestones
Received: 22 November 2008
Revised: 17 November 2009
Accepted: 29 November 2009
Published: 27 December 2009
Authors
Juan Galvis
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
United States
http://www.math.tamu.edu/~jugal
Marcus Sarkis
Worcester Polytechnic Institute
Mathematical Sciences Department
100 Institute Road
Worcester, MA 01609
United States
Instituto Nacional de Matemática Pura e Aplicada
Estrada Dona Castorina 110
22460-320 Rio de Janeiro, RJ
Brazil
http://www.wpi.edu/~msarkis