language: en

Space-time domain decomposition for parabolic problems

Eldar Giladi
Applied Mathematics 217-50 Caltech,
Pasadena CA 91125.
Herbert B. Keller
Applied Mathematics 217-50 Caltech,
Pasadena CA 91125.
March 25, 1997

Abstract We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The asymptotic convergence rate is superlinear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However, it is independent of the number of subdomains. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. In two space dimensions, the iteration possesses the smoothing property: high modes of the error are damped much faster then low modes. This is a result of the natural smoothing property of the heat equation. Numerical calculations illustrate our analysis.

1 Introduction