Explicit Determination of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators

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

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

The Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along with well-posedness, it offers a good balance between convergence rate, computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear SchrödingerSchr¨Schrödinger equation. Some illustrating numerical experiments are proposed for the one-and two-dimensional problems.

Mots clés: dynamics; Robin boundary condition; fast convergence; Optimized Schwarz Waveform Relaxation; domain decomposition method; pseudodifferential operators; SchrödingerSchr¨Schrödinger equation; stationary states

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

Xavier Antoine, Emmanuel Lorin. Explicit Determination of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators. Communications in Computational Physics, 2020, 27, pp.1032-1052. ⟨10.4208/cicp.OA-2018-0259⟩. ⟨hal-01929066⟩