We consider a differential system based on the coupling of the Navier-Stokes and Darcy equations for modeling the interaction between surface and porousmedia flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov-Poincaré operators, and we prove its wellposedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.

This paper derives a linear convergence for the Schwarz overlapping domain decomposition method when applied to constrained minimization problems. The convergence analysis is based on a minimization approach to the corresponding functional over a convex set. A general framework of convergence is established for some multiplicative Schwarz algorithm. The abstract theory is particularly applied to some obstacle problems, which yields a linear convergence for the corresponding Schwarz overlapping domain decomposition method of one and two levels. Numerical experiments are presented to confirm the convergence estimate derived in this paper.

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The fluctuation of mechanical fields arising in polycrystals is investigated. These materials are viewed as composites of the Hashin-Shtrikman type with a large number of anisotropic phases and a 'granular' topology. We show that the estimation of the intra-phase stress and strain (rate) second moments comes down to the resolution of a linear system of equations. Applied to a linear viscous face-centred cubic (FCC) polycrystal, it is observed that significant local slip rates are estimated even when the corresponding Schmid factor vanishes, due to the intergranular interactions. For the application to viscoplastic polycrystals, the secant and affine nonlinear extensions of the self-consistent scheme are compared. At large stress sensitivity (n = 30), it is observed that the secant linearization leads to almost uniform slip rates for all slip systems in every phase, whereas the affine approach predicts a much larger spread. Furthermore, there is no one-to-one relation between the phase-average stress (or strain rate) and the corresponding second moment. It is emphasized that intra-phase strain-rate heterogeneities should be accounted for when dealing with microstructure evolution.