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

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 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.

Asymptotic Convergence Rates of Schwarz Waveform Relaxation Algorithms for Schrödinger Equations with an Arbitrary Number of Subdomains

“…We are interested in this paper in the analysis of the rate of convergence of some SWR Domain Decomposition Methods (DDMs) by using an arbitrary number of subdomains. This study is an extension of existing results about the convergence of SWR algorithms on two subdomains [8][9][10]. We show that the convergence rates established for two subdomains are actually still accurate estimates for an arbitrary number of sufficiently large subdomains and bounded potentials.…”

“…We study the convergence of the SWR methods by using an arbitrary number m ≥ 2 of subdomains, for computing the point spectrum of the Schrödinger Hamiltonian by using the imaginary-time method [3,[5][6][7][12][13][14][15]. We refer to [8,18] regarding the well-posedness of the equation and of the convergence of the algorithms. We intend to determine the ground state to the following one-dimensional Schrödinger Hamitonian − △ +V(x) (used in quantum mechanics, acoustics wave propagation, optics), where the potential V is supposed to be smooth and bounded with bounded derivative.…”

“…For linear advection and diffusion reaction equations, we refer to [18]. The analysis and derivation for the Schrödinger equation in the timedependent case is presented in [10,11,16,17], while the stationary equation is studied in [8,9].…”

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

“…This paper is precisely dedicated to this question for the Schrödinger equation in the one-dimensional case with non-constant potentials. The strategy which is proposed can in principle, be applied in higher dimension and in the stationary case, see [10,11]. It extensively uses pseudo-differential calculus [2,24,29], and was originally developed for deriving and analyzing high-order absorbing boundary conditions for classical and quantum wave equations, as well as diffusion equations [7,8,13,16,22].…”

Domain decomposition method for the N-body time-independent and time-dependent Schrödinger equations

“…reach convergence, as described in Section 4. Notice that k (cvg) is strongly dependent on the type of transmission conditions [7]. We get Assuming that β (S) p , (resp.…”