Convergence Analysis of a Domain Decomposition FEM Approximation of the Isentropic Euler Equation

We analyze an interation-by-subdomain algorithm of Dirichlet\Dirichlet type for the isentropic Euler equation. Focusing on subsonic flows, which are the ones showing the most interesting features in a domain decomposition framework. The main attention is paid to the spatial decomposition, and the problem is advanced in time by means of a semi-implicit Euler scheme. We enforce the continuity on the interface of the inviscid flux, and, in the one-dimentional case, we prove convergence of the algorithm in characteristic variables for both the semi-discrete problem and the fully discrete one, where the equation is discretized in space via Streamline Diffusion Finite Elements. In both cases, the interface mapping is showed to be a contraction: in the semi discrete case, for any choice of the time step Dt, with constant of order e (-c/Dt) (c>0), in the fully discrete case, provided the entries of the stabilizing matrix are sufficiently small. Finally, some error estimates of energy type are given.