# Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations

• M. Discacciati
• A. Quarteroni
Conference paper

## Summary

We introduce a differential system based on the coupling of the (Navier) Stokes equations and the Darcy equation for the modelling of the interaction between surface and subsurface flows. We formulate the problem as an interface problem and analyze the associated Steklov-Poincaré operator. We then propose a way of solving the coupled problem iteratively, based on a suitable splitting of the interface conditions, allowing the solution of two subproblems at each step.

## Keywords

Porous Medium Interface Condition Domain Decomposition Stokes Problem Couple Problem

## References

1. [1]
Bear, J. (1979): Hydraulics of groundwater. McGraw-Hill, New YorkGoogle Scholar
2. [2]
Brezzi, F. (1974): On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle 8, 129–151
3. [3]
Brezzi, F., Fortin, M. (1991): Mixed and hybrid finite element methods. Springer, New York
4. [4]
Discacciati, M., Miglio, E., Quarteroni, A. (2002): Mathematical and numerical models for coupling surface and groundwater flows. Appl. Num. Math. 43, 57–74
5. [5]
Fortin, M. (1993): Finite element solution of the Navier-Stokes equations. In: Iserles, A. (ed.) Acta numerica. 1993 Sér, Rouge. Cambridge University Press, Cambridge, pp. 239–284Google Scholar
6. [6]
Girault, V., Raviart, P.-A. (1979): Finite clement approximation of the Navier-Stokes equations. (Lecture Notes in Mathematics, vol. 749) Springer, Berlin
7. [7]
Jäger, W., Mikelić, A. (1996): On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23, 403–465
8. [8]
Jäger, W., Mikelić, A. (2000): On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127
9. [9]
Lions, J.-L., Magenes, E. (1968): Problèmes aux limites non homogènes et applications. Vol. 1. Dunod, Paris
10. [10]
Marini, L.D., Quarteroni, A. (1989): A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55, 575–598
11. [11]
Payne, L.E., Straughan, B. (1998): Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions. J. Math. Pures Appl. (9) 77, 317–354
12. [12]
Quarteroni, A., Valli, A. (1994): Numerical approximation of partial differential equations. Springer, Berlin
13. [13]
Quarteroni, A., Valli, A. (1999): Domain decomposition methods for partial differential equations. Oxford University Press, Oxford
14. [14]
Wood, W.L. (1993): Introduction to numerical methods for water resources. Oxford University Press, Oxford