DISCRETE-TIME ANALYSIS OF OPTIMIZED SCHWARZ WAVEFORM RELAXATION WITH ROBIN PARAMETERS DEPENDING ON THE TARGETED ITERATION COUNT

Arthur Arnoult (1) , Caroline Japhet (1) , Pascal Omnes (1, 2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 SGLS - Service de Génie Logiciel pour la Simulation

We propose a new approach to analyze the convergence of optimized Schwarz waveform relaxation (OSWR) iterations for parabolic problems. Departing from traditional Fourier analysis in the time direction, we explicitly solve the equations obtained using the backward Euler scheme in time, and deduce convergence properties from this solution, in the two subdomains case. Convergence is proven for any positive Robin parameter (or couple of such parameters in the two-sided case). We also show that, for any fixed value of the number of time steps, the convergence depends on a single parameter which is a combination of the diffusion coefficient, the time step and the Robin parameter. A convergence result in a finite number of iterations is also proven for a well-chosen value of the Robin parameters. This approach allows us to define efficient optimized Robin parameters that depend on the number of iterations one wishes to perform, and to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. Numerical experiments illustrate the performance of such iteration-dependent optimized Robin parameters, compared to the observed optimal ones (which also depend on the number of iterations performed). A comparison is also given with optimized parameters derived by classical Fourier transform analysis on the continuous problem (which are independent of the iterations).

Mots clés: Discrete-time convergence analysis; Iteration-dependent Robin parameters; Optimized Schwarz waveform relaxation; Robin transmission conditions

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

Arthur Arnoult, Caroline Japhet, Pascal Omnes. DISCRETE-TIME ANALYSIS OF OPTIMIZED SCHWARZ WAVEFORM RELAXATION WITH ROBIN PARAMETERS DEPENDING ON THE TARGETED ITERATION COUNT. ESAIM: Mathematical Modelling and Numerical Analysis, 2023, ⟨10.1051/m2an/2023051⟩. ⟨hal-03746438v2⟩