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International audienceOptimized Schwarz methods (OSM) are very popular methods which were introduced by P.L. Lions in [27] for elliptic problems and by B. Després in [8] for propagative wave phenomena. We give here a theory for Lions' algorithm that is the genuine counterpart of the theory developed over the years for the Schwarz algorithm. The first step is to introduce a symmetric variant of the ORAS (Optimized Restricted Additive Schwarz) algorithm [37] that is suitable for the analysis of a two-level method. Then we build a coarse space for which the convergence rate of the two-level method is guaranteed regardless of the regularity of the coefficients. We show scalability results for thousands of cores for nearly incompressible elasticity and the Stokes systems with a continuous discretization of the pressure

An Additive Schwarz Method Type Theory for Lions's Algorithm and a Symmetrized Optimized Restricted Additive Schwarz Method