Convergence analysis of additive Schwarz for the Euler equations

In a previous paper [Internat. J. Numer. Math. Fluids 37 (2001) 625], we reported on numerical experiments with a non-overlapping domain decomposition method that has been specifically designed for the calculation of steady compressible inviscid flows governed by the two-dimensional Euler equations. In the present work, we study this method from the theoretical point of view. The proposed method relies on the formulation of an additive Schwarz algorithm which involves interface conditions that are Dirichlet conditions for the characteristic variables corresponding to incoming waves (often referred to as natural or classical interface conditions), thus taking into account the hyperbolic nature of the Euler equations. In the first part of this paper, the convergence of the additive Schwarz algorithm is analyzed in the two- and three-dimensional continuous cases by considering the linearized equations and applying a Fourier analysis. We limit ourselves to the cases of two and three-subdomain decompositions with or without overlap and we obtain analytical expressions of the convergence rate of the Schwarz algorithm. Besides the fact that the algorithm is always convergent, surprisingly, there exist flow conditions for which the asymptotic convergence rate is equal to zero. Moreover, this behavior is independent of the space dimension. In the second part, we study the discrete counterpart of the non-overlapping additive Schwarz algorithm based on the implementation adopted in [Internat. J. Numer. Math. Fluids 37 (2001) 625] but assuming a finite volume formulation on a quadrangular mesh. We find out that the expression of the convergence rate is actually more characteristic of an overlapping additive Schwarz algorithm. We conclude by presenting numerical results that confirm qualitatively the convergence behavior found analytically.