Multilevel Space-Time Additive Schwarz Methods for Parabolic Equations

Abstract

In this paper, we present a multilevel space-time additive Schwarz method for solving linear system of equations arising from the discretization of parabolic equations. With this method, the problem is solved in parallel on both space and time dimensions. After establishing two important properties of the space and time decomposition, i.e., a strengthened Cauchy--Schwarz-type inequality and a stable multilevel decomposition under a space-time energy norm, we develop an optimal convergence theory in $R^2$ and $R^3$ and show how the convergence rate depends on the mesh sizes, the number of subdomains, the window size, and the number of levels. Numerical experiments carried out on a parallel computer with thousands of processors for two- and three-dimensional problems confirm the theory in terms of the number of iterations, as well as the strong and weak scalabilities. Furthermore, a detailed comparison shows that the space-time method outperforms the traditional time stepping method, parallelized only in space, when the number of processors is large.