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Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates
Christophe Besse1; Feng Xing2
1 Institut de Mathématiques de Toulouse UMR5219, Université de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose 06108 Nice Cedex 02, France, and BRGM Orl�ans France
The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 277-300.

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose–Einstein condensates (Gross– Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.

Published online:
DOI: 10.5802/smai-jcm.17
Classification: 35Q55, 65M55, 65Y05, 65M60
Keywords: nonlinear Schrödinger equation, rotating Bose–Einstein condensate, optimized Schwarz method, preconditioned algorithm, parallel algorithm

Christophe Besse 1; Feng Xing 2

1 Institut de Mathématiques de Toulouse UMR5219, Université de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France
2 Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, University Nice Sophia Antipolis, team COFFEE, INRIA Sophia Antipolis Méditerranée, Parc Valrose 06108 Nice Cedex 02, France, and BRGM Orl�ans France 
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     title = {Domain decomposition algorithms for the two dimensional nonlinear {Schr\"odinger} equation and simulation of {Bose{\textendash}Einstein} condensates},
     journal = {The SMAI Journal of computational mathematics},
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     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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Christophe Besse; Feng Xing. Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 277-300. doi : 10.5802/smai-jcm.17. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.17/

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