Abstract
In domain decomposition methods, coarse spaces are traditionally added to make the method scalable. Coarse spaces can however do much more: they can act on other error components that the subdomain iteration has difficulties with, and thus accelerate the overall solution process. We identify here the optimal coarse space for RAS, where optimal does not refer to scalable, but to best possible. This coarse space leads to convergence of the subdomain iterative method in two steps. Since this coarse space is very rich, we propose an approximation which turns out to be also very effective for multiscale problems.
Keywords
- Iteration Count
- Domain Decomposition Method
- Schwarz Method
- Coarse Space
- Restriction Matrice
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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J. Aarnes, T.Y. Hou, Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Math. Appl. Sin. Engl. Ser. 18 (1), 63–76 (2002)
X.-C. Cai, M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21 (2), 792–797 (1999)
V. Dolean, F. Nataf, R. Scheichl, N. Spillane, Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. Comput. Methods Appl. Math. 12 (4), 391–414 (2012)
Y. Efendiev, J. Galvis, R. Lazarov, J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46 (5), 1175–1199 (2012)
E. Efstathiou, M.J. Gander, Why restricted additive Schwarz converges faster than additive Schwarz. BIT 43, 945–959 (2003)
J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8 (4), 1461–1483 (2010a)
J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8 (5), 1621–1644 (2010b)
M.J. Gander, Schwarz methods over the course of time. Electron. Trans. Numer. Anal. 31, 228–255 (2008)
M.J. Gander, L. Halpern, Méthodes de décomposition de domaine. Encyclopédie électronique pour les ingénieurs, (2012)
M.J. Gander, L. Halpern, K. Santugini, Discontinuous coarse spaces for DD-methods with discontinuous iterates, in Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering (Springer, Cham, 2014a), pp. 607–616
M.J. Gander, L. Halpern, K. Santugini, A new coarse grid correction for RAS/AS, in Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering (Springer, Cham, 2014b), pp. 275–284
M.J. Gander, F. Kwok, Optimal interface conditions for an arbitrary decomposition into subdomains, in Domain Decomposition Methods in Science and Engineering XIX (Springer, Berlin, 2011), pp. 101–108
M.J. Gander, A. Loneland, T. Rahman, Analysis of a new harmonically enriched multiscale coarse space for domain decomposition methods. arXiv preprint arXiv:1512.05285 (2015)
I.G. Graham, P.O. Lechner, R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math. 106 (4), 589–626 (2007)
A. Klawonn, P. Radtke, O. Rheinbach, FETI-DP methods with an adaptive coarse space. SIAM J. Numer. Anal. 53 (1), 297–320 (2015)
P.-L. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Paris, France, 1988, pp. 1–42
J. Mandel, B. Sousedík, Adaptive coarse space selection in the BDDC and the FETI-DP iterative substructuring methods: optimal face degrees of freedom, in Domain Decomposition Methods in Science and Engineering XVI (Springer, Berlin, 2007), pp. 421–428
R. Scheichl, Robust coarsening in multiscale PDEs, in Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol. 91 (Springer, Berlin, 2013), pp. 51–62
N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126 (4), 741–770 (2014)
A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)
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Gander, M.J., Loneland, A. (2017). SHEM: An Optimal Coarse Space for RAS and Its Multiscale Approximation. In: Lee, CO., et al. Domain Decomposition Methods in Science and Engineering XXIII. Lecture Notes in Computational Science and Engineering, vol 116. Springer, Cham. https://doi.org/10.1007/978-3-319-52389-7_32
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DOI: https://doi.org/10.1007/978-3-319-52389-7_32
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