Geodesic
Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves
Antoine, Xavier1  ; Hou, Fengji2 ; Lorin, Emmanuel3
1 Institut Elie Cartan de Lorraine, Université de Lorraine, Inria Nancy-Grand Est, 54506 Vandoeuvre-lès-Nancy Cedex, France
2 School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6
3 Centre de Recherches Mathématiques, Université de Montréal Montréal, Canada, H3T1J4
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1569-1596

Voir la notice de l'article provenant de la source Numdam

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017048
Classification : 65M12, 65M06, 65M55, 74G65
Keywords: Schwarz waveform relaxation, domain decomposition method, convergence rate, Schrödinger equation, Gross-Pitaevskii equation, absorbing boundary conditions, pseudodifferential operator theory, symbolical asymptotic expansion, stationary states, imaginary-time, normalized gradient flow

Antoine, Xavier 1 ; Hou, Fengji 2 ; Lorin, Emmanuel 3

1 Institut Elie Cartan de Lorraine, Université de Lorraine, Inria Nancy-Grand Est, 54506 Vandoeuvre-lès-Nancy Cedex, France
2 School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6
3 Centre de Recherches Mathématiques, Université de Montréal Montréal, Canada, H3T1J4 
@article{M2AN_2018__52_4_1569_0,
     author = {Antoine, Xavier and Hou, Fengji and Lorin, Emmanuel},
     title = {Asymptotic estimates of the convergence of classical {Schwarz} waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1569--1596},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {4},
     year = {2018},
     doi = {10.1051/m2an/2017048},
     zbl = {1407.65152},
     mrnumber = {3878604},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017048/}
}
TY  - JOUR
AU  - Antoine, Xavier
AU  - Hou, Fengji
AU  - Lorin, Emmanuel
TI  - Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 1569
EP  - 1596
VL  - 52
IS  - 4
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017048/
DO  - 10.1051/m2an/2017048
LA  - en
ID  - M2AN_2018__52_4_1569_0
ER  - 
%0 Journal Article
%A Antoine, Xavier
%A Hou, Fengji
%A Lorin, Emmanuel
%T Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 1569-1596
%V 52
%N 4
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017048/
%R 10.1051/m2an/2017048
%G en
%F M2AN_2018__52_4_1569_0
Antoine, Xavier; Hou, Fengji; Lorin, Emmanuel. Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1569-1596. doi: 10.1051/m2an/2017048

Cité par Sources :