no. 13-14
Partial Differential Equations/Numerical Analysis
Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain
[Méthode de Schwarz pour l'équation de la chaleur non linéaire dans un domaine cylindrique]

Présenté par : Pierre-Louis Lions

Tran, Minh-Binh1
1 LAGA, UMR 7539 of CNRS, institut Galilée, université Paris XIII, 93430 Villetaneuse, France
Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 795-799.

Nous proposons dans cette Note une preuve d'existence et de convergence de l'algorithme de Schwarz pour l'équation de la chaleur non linéaire dans un domaine cylindrique. Cette preuve repose sur l'utilisation du théorème de Banach dans un espace bien choisi, et sur de nouvelles estimations d'erreur cylindriques.

We present here a proof of well-posedness and convergence for the parallel Schwarz waveform relaxation algorithm adapted to the semilinear heat equation in a cylindrical domain. It relies on a careful estimate of a local time of existence thanks to the Banach theorem in a well chosen metric space, together with new cylindrical error estimates.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.06.004
Tran, Minh-Binh 1

1 LAGA, UMR 7539 of CNRS, institut Galilée, université Paris XIII, 93430 Villetaneuse, France 
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Tran, Minh-Binh. Parallel Schwarz waveform relaxation method for a semilinear heat equation in a cylindrical domain. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 795-799. doi : 10.1016/j.crma.2010.06.004. http://www.numdam.org/articles/10.1016/j.crma.2010.06.004/

[1] Burrage, K.; Dyke, C.; Pohl, B. On the performance of parallel waveform relaxations for differential systems, Appl. Numer. Math., Volume 20 (1996), pp. 39-55

[2] Friedman, A. Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 1964

[3] Gander, M.J. A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Czech–US Workshop in Iterative Methods and Parallel Computing, Part 2, Milovy, 1997 (Numer. Linear Algebra Appl.), Volume 6 (1999) no. 2, pp. 125-145

[4] Gander, M.J.; Stuart, A.M. Space time continuous analysis of waveform relaxation for the heat equation, SIAM J., Volume 19 (1998), pp. 2014-2031

[5] Giladi, E.; Keller, H.B. Space time domain decomposition for parabolic problems, Numer. Math., Volume 93 (2002) no. 2, pp. 279-313

[6] Lieberman, G.M. Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996

[7] Lions, P.-L. On the Schwarz alternating method I (Glowinski, R.; Golub, G.H.; Meurant, G.A.; Périaux, J., eds.), First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1988, pp. 1-42

[8] Lui, S.H. On monotone iteration and Schwarz methods for nonlinear parabolic PDEs, J. Comput. Appl. Math. (2003)

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