Multilevel space-time multiplicative Schwarz preconditioner for parabolic equations

Space-time methods are able to solve time-dependent problems faster by exploiting the full power of high-performance computer. In this article, a multilevel space-time multiplicative Schwarz method is presented for solving parabolic equations in parallel on both spatial and temporal directions. In the implementation, the proposed Schwarz method is treated as preconditioner for GMRES, that is, a coupled system arising from the discretization of the parabolic equation is solved by using a multiplicative Schwarz preconditioned GMRES algorithm. We develop an optimal convergence theory to show that the convergence rate is bounded and independent of the spatial mesh sizes, the time step size, the number of subdomains, the number of levels, and the window size. Some numerical results obtained on a parallel computer with thousands of processors are presented to confirm the theory in terms of optimality and scalability. Moreover, numerical comparisons with traditional time-stepping algorithms show that the proposed space-time method earns lots of benefits when the number of processor cores is large.