The behavior of domain decomposition methods when the overlapping length is large

Minh-Binh Tran

Open Mathematics (2014)

  • Volume: 12, Issue: 10, page 1602-1614
  • ISSN: 2391-5455

Abstract

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In this paper, we introduce a new approach for the convergence problem of optimized Schwarz methods by studying a generalization of these methods for a semilinear elliptic equation. We study the behavior of the algorithm when the overlapping length is large.

How to cite

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Minh-Binh Tran. "The behavior of domain decomposition methods when the overlapping length is large." Open Mathematics 12.10 (2014): 1602-1614. <http://eudml.org/doc/269075>.

@article{Minh2014,
abstract = {In this paper, we introduce a new approach for the convergence problem of optimized Schwarz methods by studying a generalization of these methods for a semilinear elliptic equation. We study the behavior of the algorithm when the overlapping length is large.},
author = {Minh-Binh Tran},
journal = {Open Mathematics},
keywords = {Domain decomposition; Schwarz methods; Semilinear elliptic equations; domain decomposition; semilinear elliptic equations; convergence; algorithm},
language = {eng},
number = {10},
pages = {1602-1614},
title = {The behavior of domain decomposition methods when the overlapping length is large},
url = {http://eudml.org/doc/269075},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Minh-Binh Tran
TI - The behavior of domain decomposition methods when the overlapping length is large
JO - Open Mathematics
PY - 2014
VL - 12
IS - 10
SP - 1602
EP - 1614
AB - In this paper, we introduce a new approach for the convergence problem of optimized Schwarz methods by studying a generalization of these methods for a semilinear elliptic equation. We study the behavior of the algorithm when the overlapping length is large.
LA - eng
KW - Domain decomposition; Schwarz methods; Semilinear elliptic equations; domain decomposition; semilinear elliptic equations; convergence; algorithm
UR - http://eudml.org/doc/269075
ER -

References

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  1. [1] R. A. Adams. Sobolev Spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. Zbl0314.46030
  2. [2] D. Bennequin, M. J. Gander, and L. Halpern. A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comp., 78(265):185–223, 2009. http://dx.doi.org/10.1090/S0025-5718-08-02145-5 Zbl1198.65179
  3. [3] Filipa Caetano, Martin J. Gander, Laurence Halpern, and Jérémie Szeftel. Schwarz waveform relaxation algorithms with nonlinear transmission conditions for reaction-diffusion equations. In Domain decomposition methods in science and engineering XIX, volume 78 of Lect. Notes Comput. Sci. Eng., pages 245–252. Springer, Heidelberg, 2011. Zbl1217.65185
  4. [4] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. Zbl0902.35002
  5. [5] M. J. Gander. A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numer. Linear Algebra Appl., 6(2):125–145, 1999. Czech-US Workshop in Iterative Methods and Parallel Computing, Part 2 (Milovy, 1997). http://dx.doi.org/10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4 
  6. [6] M. J. Gander and L. Halpern. Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Numer. Anal., 45(2):666–697 (electronic), 2007. http://dx.doi.org/10.1137/050642137 Zbl1140.65063
  7. [7] M. J. Gander, L. Halpern, and F. Nataf. Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation. In Eleventh International Conference on Domain Decomposition Methods (London, 1998), pages 27–36 (electronic). DDM.org, Augsburg, 1999. 
  8. [8] M. J. Gander, L. Halpern, and F. Nataf. Optimized Schwarz methods. In Domain decomposition methods in sciences and engineering (Chiba, 1999), pages 15–27 (electronic). DDM.org, Augsburg, 2001. 
  9. [9] Martin J. Gander. Optimized Schwarz methods. SIAM J. Numer. Anal., 44(2):699–731 (electronic), 2006. http://dx.doi.org/10.1137/S0036142903425409 Zbl1117.65165
  10. [10] L. Halpern and J. Szeftel. Nonlinear nonoverlapping Schwarz waveform relaxation for semilinear wave propagation. Math. Comp., 78(266):865–889, 2009. http://dx.doi.org/10.1090/S0025-5718-08-02164-9 Zbl1198.65060
  11. [11] Jung-Han Kimn. A convergence theory for an overlapping Schwarz algorithm using discontinuous iterates. Numer. Math., 100(1):117–139, 2005. http://dx.doi.org/10.1007/s00211-004-0572-3 Zbl1067.65112
  12. [12] P.-L. Lions. On the Schwarz alternating method. I. In First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), pages 1–42. SIAM, Philadelphia, PA, 1988. 
  13. [13] P.-L. Lions. On the Schwarz alternating method. II. Stochastic interpretation and order properties. In Domain decomposition methods (Los Angeles, CA, 1988), pages 47–70. SIAM, Philadelphia, PA, 1989. 
  14. [14] P.-L. Lions. On the Schwarz alternating method. III. A variant for nonoverlapping subdomains. In Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), pages 202–223. SIAM, Philadelphia, PA, 1990. 
  15. [15] Sébastien Loisel and Daniel B. Szyld. On the geometric convergence of optimized Schwarz methods with applications to elliptic problems. Numer. Math., 114(4):697–728, 2010. http://dx.doi.org/10.1007/s00211-009-0261-3 Zbl1189.65275
  16. [16] S. Mizohata. The theory of partial differential equations. Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. Zbl0263.35001
  17. [17] E. Zeidler. Nonlinear functional analysis and its applications. II/B. Springer-Verlag, New York, 1990. Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron. http://dx.doi.org/10.1007/978-1-4612-0981-2 Zbl0684.47029

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