Version abrégée i and called them Poincaré-Steklov operators. Indeed, we are interested in using iterative methods to solve (0.2) since they would require to compute at each step k the application of S to a given value u k Γ . Then, owing to (0.3), this would imply to apply independently each S i , that is to solve separately the subproblems P i (u) = 0 in Ω i with suitable boundary data on the interface. In order to increase the convergence rate of the iterative method, we introduce a

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... ioner or, more generally, a scaling operator, say P. At the stage of choosing P, the analysis of the Steklov-Poincaré operators is crucial to identify a preconditioner which would be spectrally equivalent to S and, therefore, would serve to achieve convergence in a number of steps independent of the physical quantities characterizing S (or, Thesis Outline In chapter 1 we introduce the setting of the coupled Navier-Stokes/Darcy problem. In particular, we discuss the issue of matching conditions between the two subproblems and we briefly present the approach based on homogenization theory that has been used in the literature to derive them. Coupling of Surface and Subsurface Flow In this chapter we introduce the Navier-Stokes and Darcy equations that we shall extensively use in the following. In particular, we shall discuss the derivation and physical meaning of these equations and we shall address the issue of finding suitable coupling conditions to describe the filtration processes between free fluids and porous media.