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

Abstract

This paper is dedicated to the analysis of the rate of convergence of the classical and quasioptimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

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.…”

“…The principle for analyzing the rate of convergence in the timedependent equation is closely related to the stationary case (see also [10]). Basically, we have to replace t (resp. )…”

“…We introduce a mapping  (O) defined as follows We again find the same convergence rate as in [10] for two subdomains, but the overall error is still shifted in logscale, by a positive constant linearly dependent on log(m).…”

“…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].…”

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

“…The overall domain (−6.5, 6.5) is decomposed in L 2 = 25 subdomains, with Gaussian basis functions as defined in the previous section. On each subdomain a total of N 2 φ = 36 Gaussian local basis functions, with δ = 2 in (9), are used to construct the local solutions. The chosen Robin constant is µ = 10 in (19).…”