Abstract
Methods are developed for automatically constructing small coarse spaces of low dimension for domain decomposition algorithms for problems in three dimensions. These constructions use equivalence classes of nodes on the interface between the subdomains into which the domain of a given elliptic problem has been subdivided, e.g., by a mesh partitioner; these equivalence classes already play a central role in the design, analysis, and programming of many domain decomposition algorithms. The coarse space elements are well defined even for irregular subdomains, are continuous, and well suited for use in two-level or multi-level preconditioners such as overlapping Schwarz algorithms. Significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. The condition number estimates depend primarily on the Lipschitz parameters of the subdomains.
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References
X.-C. Cai, M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21(2), 792–797 (1999)
C.R. Dohrmann, O.B. Widlund, An overlapping Schwarz algorithm for almost incompressible elasticity. SIAM J. Numer. Anal. 47(4), 2897–2923 (2009)
C.R. Dohrmann, O.B. Widlund, Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. Int. J. Numer. Methods Eng. 82, 157–183 (2010)
C.R. Dohrmann, O.B. Widlund, An alternative coarse space for irregular subdomains and an overlapping Schwarz algorithm for scalar elliptic problems in the plane. SIAM J. Numer. Anal. 50(5), 2522–2537 (2012)
C.R. Dohrmann, O.B. Widlund, Lower dimensional coarse spaces for domain decomposition, in Proceedings of the 21th International Conference on Domain Decomposition Methods in Science and Engineering, ed. by J. Erhel, M. Gander, L. Halpern, G. Pichot, T. Sassi, O.B. Widlund. Lecture Notes in Computational Science and Engineering, vol. 98 (Springer, Cham, 2014), pp. 527–535
C.R. Dohrmann, O.B. Widlund, On the design of small coarse spaces for domain decomposition algorithms. SIAM J. Sci. Comput. 39(4), A1466–A1488 (2017)
C.R. Dohrmann, A. Klawonn, O.B. Widlund, A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners, in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering, ed. by U. Langer, M. Discacciati, D. Keyes, O. Widlund, W. Zulehner, no. 60. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 2007), pp. 247–254
C.R. Dohrmann, A. Klawonn, O.B. Widlund, Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)
M. Dryja, B.F. Smith, O.B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31(6), 1662–1694 (1994)
M. Dryja, M.V. Sarkis, O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72, 313–348 (1996)
A. Heinlein, A. Klawonn, O. Rheinbach, O. Widlund, Improving the parallel performance of overlapping Schwarz methods by using a smaller energy minimizing coarse space, in Proceedings of the 24rd International Conference on Domain Decomposition Methods (2018)
P.W. Jones, Quasiconformal mappings and extendability of functions in Sobolev space. Acta Math. 147(1–2), 71–88 (1981)
A. Klawonn, O.B. Widlund, Dual-Primal FETI methods for linear elasticity. Commun. Pure Appl. Math. 59(11), 1523–1572 (2006)
A. Toselli, O.B. Widlund, Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)
Acknowledgements
The work of the first author was supported in part by the National Science Foundation Grant DMS-1522736.
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
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Widlund, O.B., Dohrmann, C.R. (2018). Small Coarse Spaces for Overlapping Schwarz Algorithms with Irregular Subdomains. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_53
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DOI: https://doi.org/10.1007/978-3-319-93873-8_53
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