Discrete-time analysis of optimized Schwarz waveform relaxation with Robin parameters depending on the targeted iteration count

Abstract

We propose a new approach that provides new results in the convergence analysis of optimized Schwarz waveform relaxation (OSWR) iterations for parabolic problems, and allows to define efficient optimized Robin parameters that depend on the targeted iteration count, a property that is shared by the actual observed optimal parameters, while traditional Fourier analysis in the time direction leads to iteration independent parameters. This new approach is based on the exact resolution of the time semi-discrete error equations. It allows to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. While the general ideas may apply to an arbitrary space dimension, the analysis is first presented in the one dimensional case. Numerical experiments illustrate the performance obtained with such iteration-dependent optimized Robin parameters.

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

“…As far as space discretization is concerned, the solution of the discrete version of (8.2) remains close to (8.3) if the space discretization parameter is small enough with respect to ν ω ; since ω is in practice bounded by π ∆t , we expect that the above Fourier analysis may remain close to practical experiments if the term √ ν∆t is large enough compared to the space discretization parameter. This has indeed recently been observed for the heat equation in [2]. As far as time discretization is concerned, the inclusion of its effect in the convergence analysis of OSWR methods is a current topic of research, and is for example addressed in [15] where a Z− transform is used and in [2], where a discrete-time analysis of the OSWR method is proposed.…”

“…This has indeed recently been observed for the heat equation in [2]. As far as time discretization is concerned, the inclusion of its effect in the convergence analysis of OSWR methods is a current topic of research, and is for example addressed in [15] where a Z− transform is used and in [2], where a discrete-time analysis of the OSWR method is proposed. This issue is also addressed in Section 9.2.…”

“…We first follow this standard approach in this work, but in a second step modify it to also include the effect of the discretization in the time direction; the Robin parameters obtained with such a modification improve the convergence rate over the standard choice in our numerical tests. Note that studying the influence of the numerical scheme over the OSWR convergence rate is a recent approach, pursued for example in [15,26,2].…”