Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Xavier Antoine (1, 2) , Fengji Hou (3) , Emmanuel Lorin (3, 4)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 IECL - Institut Élie Cartan de Lorraine
3 School of Mathematics and Statistics [Ottawa]
4 CRM - Centre de Recherches Mathématiques [Montréal]

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 variable coefficients linear 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 [12] 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.

Mots clés: Schrödinger equation Gross-Pitaevskii equation Absorbing boundary conditions Pseudodifferential; operator theory Symbolical asymptotic expansion Stationary states Imaginary-time Continuous; Schwarz Waveform Relaxation Domain decomposition method Convergence rate; Normalized GradientFlow

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Xavier Antoine, Fengji Hou, Emmanuel Lorin. 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, 2018, 52 (4), pp.1569-1596. ⟨10.1051/m2an/2017048⟩. ⟨hal-01431866⟩