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

Abstract

We derive some asymptotic estimates of the rate of convergence of Schwarz Waveform Relaxation domain decomposition methods for the Schrödinger equation when using an arbitrary number of subdomains. Hence, we justify that under certain conditions, the rates of convergence mathematically obtained for two subdomains (Antoine et al. in ESAIM M2AN,

A Schwarz waveform relaxation method for time-dependent space fractional Schrödinger/heat equations

“…is the full real-valued symmetric matrix, A is the full positive definite (SPD) J × J matrix with coefficients deduced from (7) for the fractional Laplacian, and V is the diagonal matrix part representing the potential. More specifically, the entries of A are defined as follows:…”

“…where A V is the full real matrix deduced from (7) removing the i-term, and u n+1 2,J u n 2,J . It is of course possible to use a standard backward Euler scheme in time following…”

“…In the Classical SWR (CSWR) case, B ± is simply the identity operator, B ± = ±∂ x + γId (γ ∈ R * + ) for the Robin SWR, and B ± is a nonlocal Dirichlet-to-Neumann-like (DtN) pseudodifferential operator for the Optimal SWR (OSWR). We refer to [4,5,6,7,8,16,30] for further reading for the case of the Schrödinger equation.…”

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

“…real-) time. We refer to [8] for details about the convergence of SWR methods with an arbitrary number of subdomains. The selection of the Robin parameters λ ± ξ ± i is based on the exact same strategy as in Proposition 2.1.…”

Schwarz Waveform Relaxation Physics-Informed Neural Networks for Solving Advection-Diffusion-Reaction Equations

“…We will focus in this paper on a simple diffusion-advection-reaction equation. Still, the proposed strategy applies to any other evolution (in particular wave-like) equations, for which the convergence of SWR is proven [1,2,3,4,5,6,7,8,9,10,11]. We derive a combined DDM-SWR and Physics-Informed Neural Network (PINN) method for solving local and nonlocal diffusion-advection-reaction equations.…”