Overlapping optimized Schwarz methods for parabolic equations in 𝑛 dimensions

Tran, Minh-Binh

doi:10.1090/S0002-9939-2012-11522-9

Overlapping optimized Schwarz methods for parabolic equations in $n$ dimensions
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Proc. Amer. Math. Soc. 141 (2013), 1627-1640 Request permission

Abstract:

We introduce in this paper a new tool to prove the convergence of the overlapping optimized Schwarz methods with multisubdomains. The technique is based on some estimates of the errors on the boundaries of the overlapping strips. Our guiding example is an $n$-dimensional linear parabolic equation.

References

Jean-David Benamou and Bruno Desprès, A domain decomposition method for the Helmholtz equation and related optimal control problems, J. Comput. Phys. 136 (1997), no. 1, 68–82. MR 1468624, DOI 10.1006/jcph.1997.5742
D. Bennequin, M. J. Gander, and L. Halpern, A homographic best approximation problem with application to optimized Schwarz waveform relaxation, Math. Comp. 78 (2009), no. 265, 185–223. MR 2448703, DOI 10.1090/S0025-5718-08-02145-5
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
M. J. Gander and L. Halpern, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems, SIAM J. Numer. Anal. 45 (2007), no. 2, 666–697. MR 2300292, DOI 10.1137/050642137
M. J. Gander, L. Halpern, and F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, Eleventh International Conference on Domain Decomposition Methods (London, 1998) DDM.org, Augsburg, 1999, pp. 27–36. MR 1827406
Martin J. Gander, Laurence Halpern, and Frederic Nataf, Optimized Schwarz methods, Domain decomposition methods in sciences and engineering (Chiba, 1999) DDM.org, Augsburg, 2001, pp. 15–27. MR 1827519
Loic Gouarin, Python module Optimism. Personal communication.
Laurence Halpern, Caroline Japhet, and Jérémie Szeftel, Discontinuous Galerkin and nonconforming in time optimized Schwarz waveform relaxation, Domain decomposition methods in science and engineering XIX, Lect. Notes Comput. Sci. Eng., vol. 78, Springer, Heidelberg, 2011, pp. 133–140. MR 2867652, DOI 10.1007/978-3-642-11304-8_{1}3
Laurence Halpern and Jérémie Szeftel, Nonlinear nonoverlapping Schwarz waveform relaxation for semilinear wave propgation, Math. Comp. 78 (2009), no. 266, 865–889. MR 2476563, DOI 10.1090/S0025-5718-08-02164-9
C. Japhet and Frédéric Nataf, The best interface conditions for domain decomposition methods: absorbing boundary conditions, Absorbing boundaries and layers, domain decomposition methods, Nova Sci. Publ., Huntington, NY, 2001, pp. 348–373. MR 2039948
P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510
P.-L. Lions, On the Schwarz alternating method. II. Stochastic interpretation and order properties, Domain decomposition methods (Los Angeles, CA, 1988) SIAM, Philadelphia, PA, 1989, pp. 47–70. MR 992003
P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 202–223. MR 1064345
Minh-Binh Tran. Convergence Properties of Overlapping Schwarz Domain Decomposition Algorithms. Submitted.