Implicit Space-Time Domain Decomposition Methods for Stochastic Parabolic Partial Differential Equations

We introduce and study parallel space-time domain decomposition methods for solving deterministic and stochastic parabolic equations. Traditional parallel algorithms solve parabolic problems time step by time step. The parallelism is restricted to each time step, and the algorithms are purely sequential in time. In this paper, we develop some overlapping Schwarz methods whose subdomains cover both space and time variables, and we show that the methods work well for stochastic parabolic equations discretized with an implicit stochastic Galerkin method. One-and two-level Schwarz preconditioned recycling GMRES methods are carefully investigated such that many components of the methods are reused when a large number of linear systems are solved. The key elements of this approach include an ordering algorithm and two grouping algorithms. We present some experimental results obtained on a parallel computer with more than one thousand processors. 1. Introduction. In this paper, we develop parallel implicit algorithms for solving a parabolic partial differential equation (PDE)