Coupling Parareal and Dirichlet-Neumann/Neumann-Neumann Waveform Relaxation Methods for the Heat Equation
The Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods are non-overlapping spatial domain decomposition methods to solve evolution problems, while the parareal algorithm is in time parallel fashion. Based on the combinations of space and time parallel strategies, we present and analyze parareal Dirichlet-Neumann and parareal Neumann-Neumann waveform relaxation for parabolic problems. Between these two algorithms, parareal Neumann-Neumann waveform relaxation is a space-time parallel algorithm, which increases the parallelism both in space and time. We derive for the heat equation the convergence results for both algorithms in one spatial dimension. We also illustrate our theoretical results with numerical experiments.
The first author is partially supported by the Natural Science Foundation of China (NSFC) under grant 61663043. The second author is partially supported by the Fundamental Research Funds for the Central Universities under grant G2018KY0306. We appreciate the comments of the reviewers that led to a better presentation.