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2019 | OriginalPaper | Chapter

8. Numerical Results

Author : Ulrich Wilbrandt

Published in: Stokes–Darcy Equations

Publisher: Springer International Publishing

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Abstract

The algorithms described in Chap. 7 are implemented in the C++ finite element code ParMooN (Wilbrandt et al., Comput Math Appl 74(1):74–88, 2017). In this chapter several examples from the literature are introduced and numerical results shown. To begin with, a more general discussion on numerical examples is given.

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previous chapter Algorithms
Footnotes
1
This is occasionally called direct numerical simulation (DNS).
 
2
In fact it is also possible to have a right-hand side g ∈ L 2(Ω) in the equation ∇⋅u = g. While the saddle point theory covers this case as well, fluids are divergence-free and so examples are expected to respect this.
 
3
Iterative solution algorithms for the Steklov–Poincaré equations (6.​16) and (6.​17) use a preconditioner which adds to the complexity of each iteration, which has to be taken into account, see also at the end of Sect. 7.​4.
 
4
The surfaces sometimes intersect (especially for larger γ p and larger K) which is not correctly visible in the figure. However the presentation is clearer without these intersections.
 
5
As a general similarity let a, b > 0. If γa + b < ε for some ε, then a < ε∕γ and b < ε. That means if a and b are equations and γ large, equation a is fulfilled with a higher accuracy compared with b.
 
Literature
[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
[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
[CW07a]
M.B. Cardenas and J.L. Wilson. Dunes, turbulent eddies, and interfacial exchange with permeable sediments. Water Resour. Res., 43:08412, 2007.
[CW07b]
M.B. Cardenas and J.L. Wilson. Hydrodynamics of coupled flow above and below a sediment–water interface with triangular bedforms. Adv. Water Resour., 30:301–313, 2007.CrossRef
[DQ09]
M. Discacciati and A. Quarteroni. Navier-Stokes/Darcy coupling: modeling, analysis and numerical approximation. Revista Matematica Complutense, 22(2):315–426, 2009.MathSciNetMATH
[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
Metadata
Title
Numerical Results
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_8

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