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

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

^* Corresponding author: xavier.antoine@univ-lorraine.fr

Received: 30 August 2016
Accepted: 25 September 2017

Abstract

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.