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.

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
Mots clés : 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
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     title = {Asymptotic estimates of the convergence of classical {Schwarz} waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves},
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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. http://www.numdam.org/articles/10.1051/m2an/2017048/

[1] M. Al. -Khaleel, A.E. Ruehli and M.J. Gander, Optimized waveform relaxation methods for longitudinal partitioning of transmission lines. IEEE Trans. Circuits Syst., 56 (2009) 1732–1743. | DOI | MR | Zbl

[2] X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184 (2013) 2621–2633. | DOI | MR | Zbl

[3] X. Antoine and C. Besse, Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrödinger equation. J. Math. Pures Appl. 80 (2001) 701–738. | DOI | MR | Zbl

[4] X. Antoine, C. Besse and S. Descombes, Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations. SIAM J. Numer. Anal. 43 (2006) 2272–2293. | DOI | MR | Zbl

[5] X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential. J. Comput. Phys. 228 (2009) 312–335. | DOI | MR | Zbl

[6] X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part I: Construction and a priori estimates. Math. Models Methods Appl. Sci. 22 (2012) 1250026. | DOI | MR | Zbl

[7] X. Antoine, C. Besse and V. Mouysset, Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math. Comput. 73 (2004) 1779–1799. | DOI | MR | Zbl

[8] X. Antoine, C. Besse and J. Szeftel, Towards accurate artificial boundary conditions for nonlinear PDEs through examples. Cubo 11 (2009) 29–48. | MR | Zbl

[9] X. Antoine and R. Duboscq, GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations I: Computation of stationary solutions. Comput. Phys. Commun. 185 (2014) 2969–2991. | DOI | Zbl

[10] X. Antoine and R. Duboscq, Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates. J. Comput. Phys. 258C (2014) 509–523. | DOI | MR | Zbl

[11] X. Antoine and R. Duboscq, Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity, in Nonlinear Optical and Atomic Systems: at the Interface of Mathematics and Physics. CEMPI Subseries, 1st. In Vol. 2146 of Lect. Notes Math. Springer (2015) 49–145. | DOI | MR | Zbl

[12] X. Antoine and E. Lorin, An analysis of schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations. Numer. Math. 137 (2017) 923–958. | DOI | MR | Zbl

[13] X. Antoine and E. Lorin, Lagrange Schwarz waveform relaxation domain decomposition methods for linear and nonlinear quantum wave problems. Appl. Math. Lett. 57 (2016) 38–45. | DOI | MR | Zbl

[14] X. Antoine, E. Lorin and A.D. Bandrauk, Domain decomposition method and high-order absorbing boundary conditions for the numerical simulation of the time dependent schrödinger equation with ionization and recombination by intense electric field. J. Sci. Comput. 64 (2015) 620–646. | DOI | MR | Zbl

[15] W. Bao, Ground states and dynamics of multicomponent Bose-Einstein condensates. Multiscale Model. Simul. 2 (2004) 210–236. | DOI | MR | Zbl

[16] W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic Relat. Models. 6 (2013) 1–135. | DOI | MR | Zbl

[17] W. Bao, I.-L. Chern and F.Y. Lim, Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates. J. Comput. Phys. 219 (2006) 836–854. | DOI | MR | Zbl

[18] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674–1697. | DOI | MR | Zbl

[19] W. Bao and W. Tang, Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional. J. Comput. Phys. 187 (2003) 230–254. | DOI | MR | Zbl

[20] D. Baye and J-M. Sparenberg, Resolution of the Gross-Pitaevskii equation with the imaginary-time method on a Lagrange mesh. Phys. Rev. E 82 (2010) 056701.

[21] C. Besse and F. Xing, Domain decomposition algorithms for two dimensional linear schrödinger equation. J. Sci. Comput. (2017) 1–26. | MR

[22] C. Besse and F. Xing, Schwarz waveform relaxation method for one-dimensional schrödinger equation with general potential. Numer. Algorithms 74 (2017) 393–426. | MR

[23] M. L. Chiofalo, S. Succi and M. P. Tosi, Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm. Phys. Rev. E 62 (2000) 7438.

[24] V. Dolean, P. Jolivet and F. Nataf, An introduction to domain decomposition methods: theory and parallel implementation (2015). | DOI | MR | Zbl

[25] M. Gander and L. Halpern, Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Num. Anal. 45 (2007). | DOI | MR | Zbl

[26] M.J. Gander, Overlapping Schwarz for linear and nonlinear parabolic problems. Procs. of the 9th International Conference on Domain Decomposition (1996) 97–104.

[27] M.J. Gander, Optimal Schwarz waveform relaxation methods for the one-dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643–1681. | DOI | MR | Zbl

[28] M.J. Gander, Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. | DOI | MR | Zbl

[29] M.J. Gander, Optimized Schwarz waveform relaxation methods for advection diffusion problems. SIAM J. Numer. Anal. (2007) 666–697. | DOI | MR | Zbl

[30] M.J. Gander, L. Halpern and F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation. DDM.org, Augsburg (1999) 27–36. | MR

[31] M.J. Gander, F. Kwok and B. Mandal, Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Preprint arXiv: (2015). | arXiv | MR

[32] L. Halpern and J. Szeftel, Optimized and quasi-optimal Schwarz waveform relaxation for the one-dimensional Schrödinger equation. Math. Models Methods Appl. Sci. 20 (2010) 2167–2199. | DOI | MR | Zbl

[33] R. Lascar, Propagation des singularités des solutions d’équations pseudo-différentielles quasi homogènes. Ann. Inst. Fourier (Grenoble) 27 (1977) vii–viii, 79–123. | DOI | Numdam | MR | Zbl

[34] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg (1972). | MR | Zbl

[35] E. Lorin, Schwarz waveform relaxation domain decomposition methodology for the N-body time-independent and time-dependent Schrödinger equation. Submitted (2017).

[36] E. Lorin, X. Yang and X. Antoine, Frozen gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime. J. Comput. Phys. 315 (2016) 221–237. | DOI | MR | Zbl

[37] B.C. Mandal, A time-dependent Dirichlet-Neumann method for the heat equation. Domain Decomposition Methods in Science and Engineering XXI. Springer International Publishing (2014) 467–475. | DOI | MR | Zbl

[38] L. Nirenberg, Lectures on linear partial differential equations. Amer. Math. Soc., Providence, R.I. (1973). | MR | Zbl

[39] C.J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases. Cambridge University Press (2002).

[40] L.P. Pitaevskii and S. Stringari, Bose-Einstein condensation, vol. 116. Clarendon press (2003). | MR | Zbl

[41] R. Zeng and Y. Zhang, Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates. Comput. Phys. Commun. 180 (2009) 854–860. | DOI | MR | Zbl

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