Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems

We present and analyze waveform relaxation variants of the Dirichlet-Neumann and Neumann- Neumann methods for parabolic problems. These methods are based on a non-overlapping spatial domain decomposition, and each iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, unlike for elliptic problems, each subdomain solve now involves a solution in space and time, and the interface conditions are also time-dependent. We show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann and Neumann-Neumann methods converge superlinearly for an optimal choice of the relaxation parameter, similar to the case of Schwarz waveform relaxation algorithms. Our analysis is based on Laplace transforms and detailed kernel estimates. The convergence rate depends on the size of the subdomains as well as the length of the time window. For any other choice of the relaxation parameter, convergence is only linear. We illustrate our results with numerical experiments.