Lagrange–Schwarz Waveform Relaxation domain decomposition methods for linear and nonlinear quantum wave problems

Xavier Antoine (1, 2, 3) , Emmanuel Lorin (4, 5)

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

A Schwarz Waveform Relaxation (SWR) algorithm is proposed to solve by Domain Decomposition Method (DDM) linear and nonlinear Schrödinger equations. The symbols of the transparent fractional transmission operators involved in Optimized Schwarz Waveform Relaxation (OSWR) algorithms are approximated by low order Lagrange polynomials to derive Lagrange-Schwarz Waveform Relaxation (LSWR) algorithms based on local transmission operators. The LSWR methods are numerically shown to be computationally efficient, leading to convergence rates almost similar to OSWR techniques.

Mots clés: Domain decomposition method; Schrödinger equation; Schwarz waveform relaxation algorithm

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

Xavier Antoine, Emmanuel Lorin. Lagrange–Schwarz Waveform Relaxation domain decomposition methods for linear and nonlinear quantum wave problems. Applied Mathematics Letters, 2016, 57, pp.38-45. ⟨10.1016/j.aml.2015.12.012⟩. ⟨hal-01244354⟩