On the Scalability of the Schwarz Method
The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 33-68.

In this article, we analyse the convergence behaviour and scalability properties of the one-level Parallel Schwarz method (PSM) for domain decomposition problems in which the boundaries of many subdomains lie in the interior of the global domain. Such problems arise, for instance, in solvation models in computational chemistry. Existing results on the scalability of the one-level PSM are limited to situations where each subdomain has access to the external boundary, and at most only two subdomains have a common overlap. We develop a systematic framework that allows us to bound the norm of the Schwarz iteration operator for domain decomposition problems in which subdomains may be completely embedded in the interior of the global domain and an arbitrary number of subdomains may have a common overlap.

Published online:
DOI: 10.5802/smai-jcm.61
Classification: 65N55, 65F10, 65N22, 35J05, 35J57
Keywords: Domain decomposition methods; Schwarz methods; chain of atoms; Laplace equation; ddCOSMO; Scalability analysis.

Gabriele Ciaramella 1; Muhammad Hassan 2; Benjamin Stamm 2

1 Fachbereich Mathematik und Statistik, Universität Konstanz, Germany
2 Center for Computational Engineering Science, Department of Mathematics, RWTH Aachen University, Germany
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gabriele Ciaramella; Muhammad Hassan; Benjamin Stamm. On the Scalability of the Schwarz Method. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 33-68. doi : 10.5802/smai-jcm.61. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.61/

[1] V. Barone; M. Cossi Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model, J. Phys. Chem. A, Volume 102 (1998) no. 11, pp. 1995-2001 | DOI

[2] E. Cancès; Y. Maday; B. Stamm Domain decomposition for implicit solvation models, J. Chem. Phys., Volume 139 (2013), 054111 | DOI

[3] F. Chaouqui; G. Ciaramella; M. J. Gander; T. Vanzan On the scalability of classical one-level domain-decomposition methods, Vietnam J. Math., Volume 46 (2018) no. 4, pp. 1053-1088 | DOI | MR | Zbl

[4] G. Ciaramella; M. J. Gander Analysis of the parallel Schwarz Method for growing chains of fixed-sized subdomains: Part I, SIAM J. Numer. Anal., Volume 55 (2017) no. 3, pp. 1330-1356 | DOI | MR | Zbl

[5] G. Ciaramella; M. J. Gander Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part II, SIAM J. Numer. Anal., Volume 56 (2018) no. 3, pp. 1498-1524 | DOI | MR | Zbl

[6] G. Ciaramella; M. J. Gander Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III, Electron. Trans. Numer. Anal., Volume 49 (2018), pp. 201-243 | MR | Zbl

[7] G. Ciaramella; M. J. Gander; L. Halpern; J. Salomon Methods of reflections: relations with Schwarz methods and classical stationary iterations, scalability and preconditioning (2018) (preprint, https://hal.archives-ouvertes.fr/hal-01930232) | Zbl

[8] G. Ciaramella; M. Hassan; B. Stamm On the scalability of the parallel Schwarz Method in One-Dimension (2019) to appear in Proceedings of the 25th International Conference on Domain Decomposition Methods (St. John’s, Canada)

[9] G. Ciaramella; R. Höfer Non-geometric convergence of the classical alternating Schwarz method (2019) to appear in Proceedings of the 25th International Conference on Domain Decomposition Methods (St. John’s, Canada)

[10] M. J. Gander Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255 | MR | Zbl

[11] M. J. Gander Does the partition of unity influence the convergence of Schwarz methods? (2019) to appear in Proceedings of the 25th International Conference on Domain Decomposition Methods (St. John’s, Canada)

[12] M. Hassan; B. Stamm An integral equation formulation of the N-body dielectric spheres problem. Part I: Numerical analysis (2019) (https://arxiv.org/abs/1902.01315)

[13] Gander M. J.; H. Zhao Overlapping Schwarz waveform relaxation for the heat equation in N dimensions, BIT Numer. Math., Volume 42 (2002) no. 4, pp. 779-795 | DOI | MR | Zbl

[14] L. V. Kantorovich; V. I. Krylov Approximate methods of higher analysis, 1958 (translated from the third Russian edition by C. D. Benster) | Zbl

[15] A. Klamt; G. Schuurmann COSMO: A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient, J. Chem. Soc., Perkin Trans. 2 (1993), pp. 799-805 | DOI

[16] E. B. Lindgren; A. J. Stace; E. Polack; Y. Maday; B. Stamm; E. Besley An integral equation approach to calculate electrostatic interactions in many-body dielectric systems, J. Comput. Phys., Volume 371 (2018), pp. 712-731 | DOI | MR | Zbl

[17] P. L. Lions, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (1988), pp. 1-42 | Zbl

[18] P. L. Lions, Second International Symposium on Domain Decomposition Methods for Partial Differential Equations (1989), pp. 47-70 | Zbl

[19] F. Lipparini; L. Lagardère; G. Scalmani; B. Stamm; E. Cancès; Y. Maday; J. Piquemal; M. J. Frisch; B. Mennucci Quantum calculations in solution for large to very large molecules: A new linear scaling QM/continuum approach, J. Phys. Chem. Lett., Volume 5 (2014) no. 6, pp. 953-958 | DOI

[20] F. Lipparini; G. Scalmani; L. Lagardère; B. Stamm; E. Cancès; Y. Maday; J. Piquemal; M. J. Frisch; B. Mennucci Quantum, classical, and hybrid QM/MM calculations in solution: General implementation of the ddCOSMO linear scaling strategy, J. Chem. Phys., Volume 141 (2014) no. 18, 184108 | DOI

[21] F. Lipparini; B. Stamm; E. Cances; Y. Maday; B. Mennucci Fast domain decomposition algorithm for continuum solvation models: Energy and first derivatives, J. Chem. Theory Comput., Volume 9 (2013) no. 8, pp. 3637-3648 | DOI

[22] T. Mathew Uniform convergence of the Schwarz alternating method for solving singularly perturbed advection-diffusion equations, SIAM J. Numer. Anal., Volume 35 (1998) no. 4, pp. 1663-1683 | DOI | MR | Zbl

[23] P. Pulay Convergence acceleration of iterative sequences. The case of SCF iteration, Chem. Phys. Lett., Volume 73 (1980) no. 2, pp. 393-398 | DOI

[24] C. Quan; B. Stamm; Y. Maday A domain decomposition method for the polarizable continuum model based on the solvent excluded surface, Math. Models Methods Appl. Sci., Volume 28 (2018) no. 7, pp. 1233-1266 | DOI | MR | Zbl

[25] A. K. Rappé; C. J. Casewit; K. S. Colwell; W. A. Goddard III; W. M. Skiff UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations, J. Am. Chem. Soc., Volume 114 (1992) no. 25, pp. 10024-10035 | DOI

[26] A. Toselli; O. Widlund Domain decomposition methods: Algorithms and theory, Springer, 2006

[27] T. N. Truong; E. V. Stefanovich A new method for incorporating solvent effect into the classical, ab initio molecular orbital and density functional theory frameworks for arbitrary shape cavity, Chem. Phys. Lett., Volume 240 (1995) no. 4, pp. 253-260 | DOI

[28] H. F. Walker; P. Ni Anderson acceleration for fixed-point iterations, SIAM J. Numer. Anal., Volume 49 (2011) no. 4, pp. 1715-1735 | DOI | MR | Zbl

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