Séance Séminaire

Séminaire ACSIOM

Design of a Schwarz coupling method for a dimensionally heterogeneous elliptic problem

When dealing with simulation of complex physical phenomena, one may have to couple several models which levels of complexity and computational cost are adapted to the local behavior of the system. In order to avoid heavy numerical simulations, one can use the most complex model only at locations where the physics makes it necessary, and the simplest ones - usually obtained after simplifications - everywhere else. Such simplifications in the models may involve a change in the geometry and the dimension of the physical domain. In that case, one deals with dimensionally heterogeneous coupling. In order to identify the main questions that we will have to face, we will present in this talk a preliminary study in which we couple a 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 efficient 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 error between a global 2-D reference solution and the 2-D coupled one. These theoretical results will be illustrated numerically. Finally we will present some first numerical results of a test case coupling a 3-D Navier-Stokes system with a 1-D shallow water model. This work is performed in the context of a collaboration with EDF R&D. Avec Manel Tayachi, IMAG, Grenoble