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2001-22

Coupling Fluid Flow with Porous Media Flow

by Layton, W.J., Schieweck, F., Yotov, I..


Series: 2001-22, Preprints

MSC:
35Q35 PDEs in connection with fluid mechanics
65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
65N15 Error bounds
76D07 Stokes and related (Oseen, etc.) flows
76S05 Flows in porous media; filtration; seepage

Abstract:
The transport of substances back and forth between
surface and ground water is a very serious problem. We study herein
the mathematical model of this setting consisting of the Stokes
equations in the fluid region coupled with Darcy's equations in the
porous medium, coupled across the interface by the Beavers-Joseph
conditions. We prove existence of weak solutions and give a complete
analysis of a finite element scheme which allows a simulation of the
coupled problem to be uncoupled into steps involving porous media and
fluid flow subproblems. This is important because there are many
``legacy'' codes available which have been optimized for uncoupled
porous media and fluid flow.

Keywords:
coupled porous media and fluid flow, Stokes and Darcy's equations, Beavers-Joseph condition, weak solutions, finite element scheme, error estimates