The method of fundamental solutions (MFS) and domain decomposition method (DDM) are employed to solve the water-wave diffraction by a thin porous vertical breakwater of semi-infinite extent. Based on the linearized theory of water waves, the problem can be reduced to a boundary value problem with degenerate boundary. In contrast to other mesh dependent numerical method, the MFS is easier and more efficient to handle degenerate boundary value problems. Various incident wave angles and porous parameters are included to validate the power of the proposed numerical scheme. The present results demonstrate the MFS is sufficiently accurate and feasible to be used to predict water-wave diffraction by a thin permeable breakwater by comparing with literature.

In this paper we certify that the same approach proposed in previous works by Chniti et al. [C. R. Acad. Sci.342 (2006) 883–886; CALCOLO45 (2008) 111–147; J. Sci. Comput.38 (2009) 207–228] can be applied to more general operators with strong heterogeneity in the coefficients. We consider here the case of reaction-diffusion problems with piecewise constant coefficients. The problem reduces to determining the coefficients of some transmission conditionsto obtain fast convergence of domain decomposition methods.After explaining the theoretical results, we explicitly compute the coefficients in the transmission boundary conditions. The numerical results presented in this paper confirm the optimality properties.

In topology optimization problems, we are often forced to deal with large-scale numericalproblems, so that the domain decomposition method occurs naturally. Consider a typicaltopology optimization problem, the minimum compliance problem of a linear isotropicelastic continuum structure, in which the constraints are the partial differentialequations of linear elasticity. We subdivide the partial differential equations into twosubproblems posed on non-overlapping sub-domains. In this paper, we consider the resultingproblem as multilevel one and show that it can be written as one level problem

The paper deals with an iterative method for numerical solving frictionlesscontact problems for two elastic bodies. Each iterative step consists of aDirichlet problem for the one body, a contact problem for the other one and twoNeumann problems to coordinate contact stresses. Convergence is proved by theBanach fixed point theorem in both continuous and discrete case. Numericalexperiments indicate scalability of the algorithm for some choices of therelaxation parameter.

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf,C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ...).