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              <p>In the present work, we study and analyze an efficient iterative coupling method for a dimensionally heterogeneous problem . We consider the case of 2-D Laplace equation with non symmetric boundary conditions with a corresponding 1-D Laplace equation. We will first show how to obtain the 1-D model from the 2-D one by integration along one direction, by analogy with the link between shallow water equations and the Navier-Stokes system. Then, we will focus on the design of an Schwarz-like iterative coupling method. We will discuss the choice of boundary conditions at coupling interfaces. We will prove the convergence of such algorithms and give some theoretical results related to the choice of the location of the coupling interface, and the control of the difference between a global 2-D reference solution and the 2-D coupled one. These theoretical results will be illustrated numerically.</p>
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              <p>Dans ce document nous étudions et analysons et une méthode de couplage multidimensionnel itérative. Nous considérons le cas de l'équation de Laplace 2-D avec des conditions aux bords non symétriques, couplée avec une équation de Laplace 1-D correspondante. dans un premier temps nous montrons comment obtenir le modèle 1-D à partir du modèle 2-D par intégration verticale et par analogie avec la dérivation des équations de Saint-Venant à partir des équations de Navier-Stokes. Ensuite nous présentons un algorithme de couplage de type Schwarz. Nous discutons le choix des conditions aux interfaces de couplage. Nous démontrons la convergence de tels algorithmes et donnons quelques résultats théoriques sur le choix de la position des interfaces de couplage. Un résultat théorique sur le contrôle de l'erreur entre la solution globale 2-D de référence et la solution 2-D couplée sera aussi donné. Enfin nous illustrons ces résultats numériquement.</p>
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