Convergence analysis of space-time domain decomposition method for parabolic equations.

Space-time domain decomposition methods have been successfully applied to solve linear systems that arise from the discretization of time-dependent problems. In this paper, we present a space-time pure multiplicative Schwarz method for solving linear parabolic equations. With this method, the systems of equations are first coupled together and then solved by preconditioned GMRES, so the solutions at arbitrary time steps can be obtained simultaneously at a time. Under some mild assumptions, we develop an optimal convergence theory and show that the convergence rate is bounded independently of the mesh sizes, the subdomain partition and the window size. Some numerical results are reported to illustrate the optimality and scalability of the proposed method. Moreover, the numerical comparison also shows that our method has a faster convergence rate.