Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation Algorithms for Parabolic Problems

TL;DR

This paper introduces waveform relaxation algorithms based on Dirichlet-Neumann and Neumann-Neumann methods for parabolic problems, demonstrating superlinear convergence on finite time intervals with optimal parameters.

Contribution

It extends classical domain decomposition methods to space-time problems using waveform relaxation, analyzing convergence with Laplace transforms and providing numerical validation.

Findings

Superlinear convergence with optimal relaxation parameter

Convergence rate depends on subdomain size and time window length

Numerical experiments confirm theoretical results

Abstract

We present a waveform relaxation version of the Dirichlet-Neumann and Neumann-Neumann methods for parabolic problems. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. For the Neumann-Neumann method, one step of the method consists of solving the subdomain problems using Dirichlet interface conditions, followed by a correction step involving Neumann interface conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using Laplace transforms, 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. 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.