## A parallel domain decomposition method for coupling of surface and groundwater flows.(English)Zbl 1229.76048

Summary: In this paper, we construct a robust parallel method based on a recently developed non-overlapping domain decomposition methodology to accurately model natural coupling of surface and groundwater flows. Stokes and Darcy equations are formulated and solved within the surface and subsurface regions, respectively. A new type of Robin-Robin boundary condition is proposed on the common boundary for the coupling of those systems. The formulation provides great flexibility for multi-physics coupling and is suitable for efficient parallel implementation. Meanwhile, it is stable with inherent system parameter variation. A numerical example is provided to verify the theory.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76D07 Stokes and related (Oseen, etc.) flows 65Y05 Parallel numerical computation 86A05 Hydrology, hydrography, oceanography
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### References:

 [1] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York · Zbl 0788.73002 [2] Ciarlet, P.G., The finite element method for elliptic problems, (1979), North-Holland Publishing Company Amsterdam · Zbl 0415.73072 [3] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. num. math., 43, 57-74, (2002) · Zbl 1023.76048 [4] Discacciati, M.; Quarteroni, A., Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, (), 3-20 · Zbl 1254.76051 [5] Discacciati, M.; Quarteroni, A., Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations, Comput. visual. sci., 6, 93-103, (2004) · Zbl 1299.76252 [6] Discacciati, M.; Quarteroni, A.; Valli, A., Robin – robin domain decomposition methods for the stokes – darcy coupling, SIAM J. numer. anal., 45, 1246-1268, (2007) · Zbl 1139.76030 [7] Galvis, J.C.; Sarkis, M., Balancing domain decomposition methods for mortar coupling stokes – darcy systems, () [8] Gartling, D.; Hickox, C.; Givler, R., Simulation of coupled viscous and porous flow problems, Compos. fluid dyn., 7, 23-48, (1996) · Zbl 0879.76104 [9] Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman Publisher Boston · Zbl 0695.35060 [10] Jager, W.; Mikelic, A., On the interface boundary condition of Beavers, Joseph and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088 [11] Jiang, B.; Bruch, J.C.; Sloss, J.M., A nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems, Numer. methods part. diff. eqn., 22, 1-17, (2006) · Zbl 1114.65072 [12] Jiang, B., Convergence analysis of P1 finite element method for free boundary problems on nonoverlapping subdomains, Comput. methods appl. mech. engrg., 196, 371-378, (2006) · Zbl 1120.76325 [13] Layton, W.L.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. numer. anal., 40, 2195-2218, (2003) · Zbl 1037.76014 [14] Lions, J.L.; Magenes, E., Non-homogeneous boundary value problems and applications, vol. I, (1972), Springer-Verlag New York · Zbl 0223.35039 [15] Mardal, K.A.; Tai, X.C.; Winther, R., A robust finite element method for darcy – stokes flow, SIAM J. numer. anal., 40, 1605-1631, (2002) · Zbl 1037.65120 [16] Quarteroni, A.; Valli, A., Domain decomposition method for partial differential equations, (1999), Oxford University Press Oxford · Zbl 0931.65118 [17] Riviere, B.; Yotov, I., Locally conservative coupling of Stokes and Darcy flows, SIAM J. numer. anal., 42, 1959-1977, (2005) · Zbl 1084.35063 [18] Salinger, A.; Aris, R.; Derby, I., Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains, Int. J. numer. methods fluids, 18, 1185-1209, (1994) · Zbl 0807.76039
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