zbMATH — the first resource for mathematics

Mathematical and numerical models for coupling surface and groundwater flows. (English) Zbl 1023.76048
Summary: We present some results on coupling Navier-Stokes with shallow water equations for surface flows, and with Darcy’s equation for groundwater flows. We discuss suitable interface conditions and show the well-posedness of the coupled problem in the case of a linear Stokes problem. An iterative method is proposed to compute the solution. At each step this method requires the solution of one problem in the fluid part and one in the porous medium. Finally, we introduce Steklov-Poincaré equation associated with the coupled problem.

76S05 Flows in porous media; filtration; seepage
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Bear, J., Hydraulics of groundwater, (1979), McGraw-Hill New York
[2] Beavers, G.S.; Joseph, D.D., Boundary conditions at a naturally permeable wall, J. fluid mech., 30, 197-207, (1967)
[3] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO anal. numér., 8, 129-151, (1974) · Zbl 0338.90047
[4] E. Campana, A. Iafrati, A domain decomposition approach for unsteady free surface flows, Preprint · Zbl 1025.76022
[5] M. Discacciati, Modelli di accoppiamento fra le equazioni di Stokes e quelle di Darcy per lo studio di problemi di idrodinamica, Degree thesis, Università degli Studi dell’Insubria, Como, Italy, 2001 (in Italian)
[6] M. Discacciati, A. Quarteroni, Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, submitted CEPFL Institut de Mathématiques, Internal Report N. 02. 2002, 2002 · Zbl 1254.76051
[7] Fontana, L.; Miglio, E.; Quarteroni, A.; Saleri, F., A finite element method for 3D hydrostatic water flows, Comput. vis. sci., 2, 2-3, 85-93, (1999) · Zbl 1067.76566
[8] 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., 23, 403-465, (1996) · Zbl 0878.76076
[9] Jäger, W.; Mikelić, A., On the interface boundary condition of Beavers, Joseph and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088
[10] Jones, I.P., Low Reynolds number flow past a porous spherical shell, Proc. camb. philos. soc., 73, 231-238, (1973) · Zbl 0262.76061
[11] Lions, J.L.; Magenes, E., Problèmes aux limites non homogènes et applications, 1, (1968), Dunod Paris
[12] E. Miglio, Mathematical and numerical modelling for enviromental applications, Ph.D. Thesis, Università degli Studi di Milano, Politecnico di Milano, Milan, Italy, 2000
[13] E. Miglio, A. Quarteroni, F. Saleri, Coupling of free surface and groundwater flows, Comput. Fluids, accepted · Zbl 1035.76051
[14] Miglio, E.; Quarteroni, A.; Saleri, F., Finite element approximation of quasi-3D shallow water equations, Comput. methods appl. mech. engrg., 174, 3-4, 355-369, (1999) · Zbl 0958.76046
[15] Nield, D.A.; Bejan, A., Convection in porous media, (1999), Springer-Verlag New York · Zbl 0924.76001
[16] Payne, L.E.; Straughan, B., 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, 4, 317-354, (1998) · Zbl 0906.35067
[17] Quarteroni, A.; Valli, A., Domain decomposition methods for partial differential equations, (1999), Oxford University Press Oxford · Zbl 0931.65118
[18] Quarteroni, A.; Valli, A., Numerical approximation of partial differential equations, (1994), Springer-Verlag Berlin · Zbl 0852.76051
[19] Wood, W.L., Introduction to numerical methods for water resources, (1993), Oxford Science Publications Oxford · Zbl 0801.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.