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

2017; Society for Industrial and Applied Mathematics; Volume: 39; Issue: 4 Linguagem: Inglês

10.1137/16m1060066

1095-7197

Ryadh Haferssas, Pierre Jolivet, F. Nataf,

Matrix Theory and Algorithms

Optimized Schwarz methods (OSMs) are very popular methods which were introduced by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), T. F. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, 1990, pp. 202--223] for elliptic problems and by B. Després in [C. R. Acad. Sci. Paris Ser. I Math., 311 (1990), pp. 313--316] for propagative wave phenomena. We give here a theory for Lions's 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 optimized restricted additive Schwarz (ORAS) algorithm [A. St-Cyr, M. J. Gander, and S. J. Thomas, SIAM J. Sci. Comput., 29 (2007), pp. 2402--2425] 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.