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2019 | OriginalPaper | Chapter

6. Stokes–Darcy Equations

Author : Ulrich Wilbrandt

Published in: Stokes–Darcy Equations

Publisher: Springer International Publishing

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Abstract

Let \(\varOmega \subset \mathbb {R}^d\) be a Lipschitz domain split into two disjoint nonempty subdomains Ω p and Ω f which are Lipschitz, too. The index p refers to the Darcy subdomain where a porous medium is modeled, while the index f refers to the free flow domain with a Stokes model.

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Footnotes
1
Sometimes also called Beavers–Joseph–Saffman–Jones condition.
 
2
With this ψ it is \(a_{\mathrm {p}}(\varphi ,\psi ) = a_{\mathrm {p}}^R(\varphi ,\psi )\).
 
3
With this v it is https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-02904-3_6/470501_1_En_6_IEq224_HTML.gif .
 
4
The signs and positions of η f and η p are chosen such that the resulting system matches that in [DQV07] and [CGHW11].
 
5
For this approach it is assumed that Λ f = Λ p, see Sect. 6.6.8.
 
Literature
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Dietrich Braess. Finite elements. Cambridge University Press, Cambridge, third edition, 2007. Theory, fast solvers, and applications in elasticity theory, Translated from the German by Larry L. Schumaker.
[BS08]
Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008.
[CGHW10]
Yanzhao Cao, Max Gunzburger, Fei Hua, and Xiaoming Wang. Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci., 8(1):1–25, 2010.MathSciNetCrossRef
[CGHW11]
Wenbin Chen, Max Gunzburger, Fei Hua, and Xiaoming Wang. A parallel robin-robin domain decomposition method for the Stokes-Darcy system. SIAM J. Numerical Analysis, 49(3):1064–1084, 2011.MathSciNetCrossRef
[CGHW14]
Yanzhao Cao, Max Gunzburger, Xiaoming He, and Xiaoming Wang. Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems. Math. Comp., 83(288):1617–1644, 2014.MathSciNetCrossRef
[Cia02]
Philippe G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].
[CJW14]
Alfonso Caiazzo, Volker John, and Ulrich Wilbrandt. On classical iterative subdomain methods for the Stokes-Darcy problem. Comput. Geosci., 18(5):711–728, 2014.MathSciNetCrossRef
[DQV07]
Marco Discacciati, Alfio Quarteroni, and Alberto Valli. Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal., 45(3):1246–1268, 2007.MathSciNetCrossRef
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Carlo D’Angelo and Paolo Zunino. Robust numerical approximation of coupled Stokes and Darcy’s flows applied to vascular hemodynamics and biochemical transport. ESAIM, Math. Model. Numer. Anal., 45(3):447–476, 2011.MathSciNetCrossRef
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Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas. Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comp., 80(276):1911–1948, 2011.MathSciNetCrossRef
[GOS11b]
Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas. Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem. Numer. Methods Partial Differential Equations, 27(3):721–748, 2011.MathSciNetCrossRef
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Juan Galvis and Marcus Sarkis. Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal., 26:350–384, 2007.MathSciNetMATH
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Willi Jäger and Andro Mikelić. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math., 60(4):1111–1127, 2000.MathSciNetCrossRef
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I.P. Jones. Low Reynolds number flow past a porous spherical shell. Math. Proc. Cambridge Philos. Soc., 73:231–238, 1973.CrossRef
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William J. Layton, Friedhelm Schieweck, and Ivan Yotov. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal., 40(6):2195–2218 (2003), 2002.MathSciNetCrossRef
[MMS15]
Antonio Márquez, Salim Meddahi, and Francisco-Javier Sayas. Strong coupling of finite element methods for the Stokes-Darcy problem. IMA J. Numer. Anal., 35(2):969–988, 2015.MathSciNetCrossRef
[RM14]
Iryna Rybak and Jim Magiera. A multiple-time-step technique for coupled free flow and porous medium systems. J. Comput. Phys., 272:327–342, 2014.MathSciNetCrossRef
[Saf71]
P.G. Saffman. On the boundary condition at the interface of a porous medium. Stud. Appl. Math., 50:93–101, 1971.CrossRef
Metadata
Title
Stokes–Darcy Equations
Author
Ulrich Wilbrandt
Copyright Year
2019
Book
Stokes–Darcy Equations

Print ISBN: 978-3-030-02903-6

Electronic ISBN: 978-3-030-02904-3

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-02904-3_6

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