Coupling Darcy and Stokes equations for porous media with cracks

Christine Bernardi; Frédéric Hecht; Olivier Pironneau

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 1, page 7-35
  • ISSN: 0764-583X

Abstract

top
In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.

How to cite

top

Bernardi, Christine, Hecht, Frédéric, and Pironneau, Olivier. "Coupling Darcy and Stokes equations for porous media with cracks." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.1 (2005): 7-35. <http://eudml.org/doc/245800>.

@article{Bernardi2005,
abstract = {In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.},
author = {Bernardi, Christine, Hecht, Frédéric, Pironneau, Olivier},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Darcy and Stokes equations; finite elements; error estimates; finite element discretization; interface pressure continuity},
language = {eng},
number = {1},
pages = {7-35},
publisher = {EDP-Sciences},
title = {Coupling Darcy and Stokes equations for porous media with cracks},
url = {http://eudml.org/doc/245800},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Bernardi, Christine
AU - Hecht, Frédéric
AU - Pironneau, Olivier
TI - Coupling Darcy and Stokes equations for porous media with cracks
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 1
SP - 7
EP - 35
AB - In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.
LA - eng
KW - Darcy and Stokes equations; finite elements; error estimates; finite element discretization; interface pressure continuity
UR - http://eudml.org/doc/245800
ER -

References

top
  1. [1] Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96 (2003) 17–42. Zbl1050.76035
  2. [2] M. Amara, D. Capatina-Papaghiuc, E. Chacón-Vera and D. Trujillo, Vorticity–velocity–pressure formulation for Navier–Stokes equations. Comput. Vis. Sci. 6 (2004) 47–52. Zbl1299.76059
  3. [3] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823–864. Zbl0914.35094
  4. [4] C. Bègue, C. Conca, F. Murat and O. Pironneau, Les équations de Stokes et de Navier–Stokes avec des conditions aux limites sur la pression. Nonlinear Partial Differ. Equ. Appl., Collège de France Seminar IX (1988) 179–264. Zbl0687.35069
  5. [5] C. Bernardi, C. Canuto and Y. Maday, Un problème variationnel abstrait. Application d’une méthode de collocation pour les équations de Stokes. C.R. Acad. Sci. Paris série I 303 (1986) 971–974. Zbl0612.49004
  6. [6] C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237–1271. Zbl0666.76055
  7. [7] S. Bertoluzza and V. Perrier, The mortar method in the wavelet context. ESAIM: M2AN 35 (2001) 647–673. Zbl0995.65131
  8. [8] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431–2444. Zbl0866.65071
  9. [9] D.-G. Calugaru, Modélisation et simulation numérique du transport de radon dans un milieu poreux fissuré ou fracturé. Problème direct et problèmes inverses comme outils d’aide à la prédiction sismique, Thesis, Université de Franche-Comté, Besançon (2002). 
  10. [10] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973) 33–76. Zbl0302.65087
  11. [11] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57–74. Zbl1023.76048
  12. [12] M. Discacciati and A. Quarteroni, Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, in Proc. of ENUMATH, F. Brezzi Ed., Springer-Verlag (to appear). Zbl1254.76051MR2360703
  13. [13] M. Discacciati and A. Quarteroni, Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Vis. Sci. 6 (2004) 93–104. Zbl1299.76252
  14. [14] F. Dubois, Vorticity–velocity–pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091–1119. Zbl1099.76049
  15. [15] F. Dubois, M. Salaün and S. Salmon, First vorticity–velocity–pressure scheme for the Stokes problem, Internal Report 356, Institut Aérotechnique, Conservatoire National des Arts et Métiers, France (2002) (submitted). Zbl1054.76047
  16. [16] P.J. Frey and P.-L. George, Maillages, applications aux éléments finis. Hermès, Paris (1999). 
  17. [17] P.-L. George and F. Hecht, Nonisotropic grids. Handbook of Grid Generation, J.F. Thompson, B.K. Soni & N.P. Weatherhill Eds., CRC Press (1998). 
  18. [18] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer–Verlag (1986). Zbl0585.65077
  19. [19] F. Hecht, Construction d’une base de fonctions P 1 non conforme à divergence nulle dans 3 . RAIRO Anal. Numér. 15 (1981) 119–150. Zbl0471.76028
  20. [20] F. Hecht and O. Pironneau, FreeFem++, see www.freefem.org. 
  21. [21] H. Kawarada, E. Baba and H. Suito, Effects of spilled oil on coastal ecosystems, in the Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering 2000, CD-ROM proceedings (2001). 
  22. [22] H. Kawarada, E. Baba and H. Suito, Effects of wave breaking action on flows in tidal-flats, in Computational Fluid Dynamics for the 21st Century, M. Hafez, K. Morinishi and J. Périaux Eds., Springer. Notes on Numerical Fluid Mechanics 78 (2001) 275–289. 
  23. [23] W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, Preprint of the University of Magdebourg, report N 22-01 (2001). Zbl1037.76014MR1974181
  24. [24] J.-C. Nedelec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315–341. Zbl0419.65069
  25. [25] R.A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349–357. Zbl0485.65049
  26. [26] P.-A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of Finite Element Methods. Springer, Berlin. Lect. Notes Math. 606 (1977) 292–315. Zbl0362.65089
  27. [27] S. Salmon, Développement numérique de la formulation tourbillon–vitesse–pression pour le problème de Stokes. Thesis, Université Pierre et Marie Curie, Paris (1999). 
  28. [28] R. Temam, Theory and Numerical Analysis of the Navier–Stokes Equations. North-Holland (1977). Zbl0383.35057
  29. [29] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). Zbl0853.65108

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.