An Additive Schwarz Method Type Theory for Lions's Algorithm and a Symmetrized Optimized Restricted Additive Schwarz Method
Abstract
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.
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Information
Published In
SIAM Journal on Scientific Computing
Pages: A1345 - A1365
DOI: 10.1137/16M1060066
ISSN (online): 1095-7197
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 4 February 2016
Accepted: 9 March 2017
Published online: 27 July 2017
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