A SUPERLINEAR CONVERGENCE ESTIMATE FOR THE PARAREAL SCHWARZ WAVEFORM RELAXATION ALGORITHM

The parareal Schwarz waveform relaxation algorithm is a new space-time parallel algorithm for the solution of evolution partial differential equations. It is based on a decomposition of the entire space-time domain both in space and in time into smaller space-time subdomains, and then computes by an iteration in parallel on all these small space-time subdomains a better and better approximation of the overall solution in space-time. The initial conditions in the space-time subdomains are updated using a parareal mechanism, while the boundary conditions are updated using Schwarz waveform relaxation techniques. A first precursor of this algorithm was presented 15 years ago, and while the method works well in practice, the convergence of the algorithm is not yet understood, and to analyze it is technically difficult. We present in this paper for the first time an accurate superlinear convergence estimate when the algorithm is applied to the heat equation. We illustrate our analysis with numerical experiments including cases not covered by the analysis, which opens up many further research directions.