A strongly conservative finite element method for the coupling of Stokes and Darcy flow

Journal of Computational Physics

Volumn 229, Issue 17, 2010, Pages 5933-5943

  Kanschat, G ^a   Riviere B ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b Rice University   (United States)

Author keywords

Beavers Joseph Saffman condition; Darcy flow; Discontinuous Galerkin; Finite element method; Mixed finite elements; Multiphysics; Stokes flow

Indexed keywords

FINITE ELEMENT METHOD; FLOW OF FLUIDS; GALERKIN METHODS; POROUS MATERIALS;

BEAVER-JOSEPH-SAFFMAN CONDITIONS; DARCY'S FLOW; DISCONTINUOUS GALERKIN; GALERKIN TECHNIQUE; INTERFACE CONDITIONS; MIXED FINITE ELEMENTS; MULTI-PHYSICS; POROUS MEDIUM FLOW; STOKES AND DARCY EQUATIONS; STOKES FLOWS;

MULTIPHYSICS;

EID: 77953614420     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.04.021     Document Type: Article

References (31)

1. Arbogast T., Brunson D. A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 2007, 11:207-218.
2. Arnold D.N. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 1982, 19(4):742-760.
3. Arnold D.N., Boffi D., Falk R.S. Approximation by quadrilateral finite elements. Math. Comput. 2002, 71:909-922.
4. Arnold D.N., Brezzi F., Cockburn B., Marini D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 2001, 39(5):1749-1779.
5. Bangerth W., Hartmann R., Kanschat G. Deal. II-a general purpose object oriented finite element library. ACM Trans. Math. Softw. 2007, 33(4).
6. Beavers G.S., Joseph D.D. Boundary conditions at a naturally impermeable wall. J. Fluid. Mech 1967, 30:197-207.
7. Brezzi F., Douglas J., Fortin M., Marini D. Efficient mixed finite element methods in two and three space variables. RAIRO Modél. Math. Anal. Numer. 1987, 21:581-604.
8. Brezzi F., Douglas J., Marini D. Two families of mixed finite elements for second order elliptic problems. Numer. Math. 1985, 47:217-235.
9. Brezzi F., Fortin M. Mixed and Hybrid Finite Element Methods 1991, Springer.
10. Burman E., Hansbo P. Stabilized Crouzeix-Raviart elements for the Darcy-Stokes problem. Numer. Methods Partial Diff. Eqn. 2005, 21:986-997.
11. Burman E., Hansbo P. A unified stabilized method for Stokes and Darcy's equations. J. Comput. Appl. Math. 2007, 198(1):35-51.
12. Cockburn B., Kanschat G., Schötzau D. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 2005, 74:1067-1095.
13. Cockburn B., Kanschat G., Schötzau D. A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 2007, 31(1-2):61-73.
14. Discacciati M., Miglio E., Quarteroni A. Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 2001, 43(1-2):57-74.
15. Gatica G., Meddahi S., Oyarzua R. A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 2009, 29:86-108.
16. Girault V., Raviart P.-A. Finite Element Approximations of the Navier-Stokes Equations 1986, Springer.
17. Girault V., Rivière B. DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 2007.
18. Hansbo P., Larson M.G. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Eng. 2002, 191:1895-1908.
19. Hanspal N., Waghode A., Nassehi V., Wakeman R. Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transport Porous Med. 2006, 64(1):1573-1634.
20. Jäger W., Mikelić A. On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1996, 23(3):403-465.
21. Jäger W., Mikelić A. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 2000, 60(4):1111-1127. (electronic).
22. Jäger W., Mikelić A. Modeling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization. Transp. Porous Med. 2009, 78(3):489-508. http://dx.doi.org/10.1007/s11242-009-9354-9.
23. G. Kanschat, Discontinuous Galerkin Methods for Viscous Flow, Deutscher Universitätsverlag, Wiesbaden, 2007.
24. Layton W.J., Schieweck F., Yotov I. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 2003, 40(6):2195-2218.
25. Nassehi V. Modelling of combined Navier-Stokes and Darcy flows in crossflow membrane filtration. Chem. Eng. Sci. 1998, 53(6):1253-1265.
26. Raviart P.-A., Thomas J.M. A mixed method for second order elliptic problems. Mathematical Aspects of the Finite Element Method 1977, 292-315. Springer, New York. I. Galligani, E. Magenes (Eds.).
27. Rivière B. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 2005, 22:479-500.
28. Rivière B., Wheeler M.F., Girault V. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 2001, 39(3):902-931.
29. Rivière B., Yotov I. Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 2005, 42:1959-1977.
30. Saffman P. On the boundary condition at the surface of a porous media. Stud. Appl. Math. 1971, 50:292-315.
31. Wang J., Ye X. New finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal. 2007, 45:1269-1286.