Abstract
We consider a differential system based on the coupling of the Navier–Stokes and Darcy equations for modeling the interaction between surface and porous-media flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov–Poincaré operators, and we prove its well-posedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.
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Adams R.: Sobolev Spaces. Academic Press, New York (1975)
Badea, L., Discacciati, M., Quarteroni, A.: Numerical analysis of the Navier–Stokes/Darcy coupling. Tech. rep., Ecole Polytechnique Fédérale de Lausanne, IACS-CMCS (2008)
Berninger, H.: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation. Ph.D. thesis, Freie Universität Berlin (2007)
Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Method. Springer, New York (1991)
Brezzi F., Rappaz J., Raviart P.: Finite dimensional approximation of nonlinear problems. Part I. Branches of nonlinear solutions. Numer. Math. 36, 1–25 (1980)
Brezzi F., Rappaz J., Raviart P.: Finite dimensional approximation of nonlinear problems. Part II. Limit points. Numer. Math. 37, 1–28 (1981)
Brezzi F., Rappaz J., Raviart P.: Finite dimensional approximation of nonlinear problems. Part III. Simple bifurcation points. Numer. Math. 38, 1–30 (1981)
Brinkman H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34 (1947)
Caloz, G., Rappaz, J.: Numerical Analysis for Nonlinear and Bifurcation Problems. In: Handbook of Numerical Analysis, vol. V, pp. 487–637. North-Holland, Amsterdam (1997)
Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2004)
Discacciati M., Miglio E., Quarteroni A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)
Discacciati, M., Quarteroni, A.: Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds.) Numerical Mathematics and Advanced Applications, ENUMATH 2001, pp. 3–20. Springer. Milan (2003)
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)
Discacciati M., Quarteroni A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22(2), 315–426 (2009)
Forchheimer P.: Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing. 45, 1782–1788 (1901)
Giorgi T.: Derivation of the Forchheimer law via matched asymptotic expansions. Transp. Porous Media 29, 191–206 (1997)
Girault V., Raviart P.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer, Berlin (1986)
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)
Jäger W., Mikelić A.: On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)
Jäger W., Mikelić A., Neuss N.: Asymptotic analysis of the laminar viscous flow over a porous bed. SIAM J. Sci. Comput. 22(6), 2006–2028 (2001)
Kantorovich, L., Akilov, G.: Functional analysis in normed spaces. Translated from the Russian by Brown, D.E., In: Robertson, A.P. (ed.) International Series of Monographs in Pure and Applied Mathematics, vol. 46. The Macmillan Co., New York (1964)
Layton W., Schieweck F., Yotov I.: Coupling fluid flow with porous media flow. SIAM J. Num. Anal. 40, 2195–2218 (2003)
Lions, J., Magenes, E.: Problèmes aux Limites Non Homogènes et Applications, vol. 1. Dunod, Paris (1968)
Marušić-Paloka E., Mikelić A.: The derivation of a nonlinear filtration law including the inertia effects via homogenization. Nonlinear Anal. 42, 97–137 (2000)
Quarteroni A., Valli A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. The Clarendon Press, Oxford University Press, New York (1999)
Yosida K.: Functional Analysis. Springer, Berlin (1974)
Zunino, P.: Mathematical and numerical modeling of mass transfer in the vascular system. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2002)
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The first author acknowledges the financial support from the program SCOPES n. IB7320–110721 and from IACS, EPFL. The other authors acknowledge the support of the FNS Project n. 200020–117587 “Interface operators and solutions algorithms for fluid–structure interaction problems with applications”. The second author acknowledges also the Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, for partially supporting this research. Finally, the third author acknowledges also the financial support from the program Cofin MIUR PRIN 2007 n. 200774A7LH_001 “Mathematical and numerical modelling for cardiovascular and fluid dynamics applications”.
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Badea, L., Discacciati, M. & Quarteroni, A. Numerical analysis of the Navier–Stokes/Darcy coupling. Numer. Math. 115, 195–227 (2010). https://doi.org/10.1007/s00211-009-0279-6
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DOI: https://doi.org/10.1007/s00211-009-0279-6