Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations

In this paper, we present a two-level space–time hybrid Schwarz preconditioner for GMRES based on the classical hybrid Schwarz method and the second-order backward differentiation formula. In the proposed method, the parabolic equations are solved in parallel on both of the space and time directions. Under some reasonable assumptions, the optimal convergence theory is developed for the proposed space–time method, i.e., the convergence rate is independent of the mesh parameters, the number of subdomains and the window size. Some numerical results are given to confirm the theory very well in terms of the convergence rate and accuracy. And the strong/weak scalability obtained with 4096 processors is also reported to show the efficiency of the proposed method.