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Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Authors: Gabriel N. Gatica, Ricardo Oyarzúa and Francisco-Javier Sayas
Journal: Math. Comp. 80 (2011), 1911-1948
MSC (2010): Primary 65N15, 65N30, 74F10, 74S05
Published electronically: February 14, 2011
MathSciNet review: 2813344
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Abstract: In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The fully-mixed concept employed here refers to the fact that we consider dual-mixed formulations in both media, which means that the main unknowns are given by the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. We apply the Fredholm and Babuška-Brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous and discrete formulations. In particular, we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates. A feasible choice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and confirming the theoretical rate of convergence, are provided.

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Additional Information

Gabriel N. Gatica
Affiliation: CI\raisebox{.75ex}2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Ricardo Oyarzúa
Affiliation: Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Francisco-Javier Sayas
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA

Received by editor(s): June 3, 2009
Received by editor(s) in revised form: July 12, 2010
Published electronically: February 14, 2011
Additional Notes: The research of the first two authors was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by CI$^{2}$MA, Universidad de Concepción, and by MECESUP project UCO 0713
The third author acknowledges support of MEC/FEDER Project MTM2007–63204 and Gobierno de Aragón (Grupo PDIE). This work was developed while the third author was at the University of Minnesota supported by a Spanish MEC grant PR2007–0016.
Article copyright: © Copyright 2011 American Mathematical Society