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

Minh-Binh Tran 1
  • 1 Basque Center for Applied Mathematics, Mazarredo 14, 48009, Bilbao, Spain

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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.

  • [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

  • [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.

  • [4] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

  • [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] 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

  • [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] 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] Martin J. Gander. Optimized Schwarz methods. SIAM J. Numer. Anal., 44(2):699–731 (electronic), 2006. http://dx.doi.org/10.1137/S0036142903425409

  • [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

  • [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

  • [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] 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] 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] 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

  • [16] S. Mizohata. The theory of partial differential equations. Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara.

  • [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

OPEN ACCESS

Journal + Issues

Search