Multilevel space-time block diagonal preconditioners for parabolic problems.

We present and investigate two new robust space–time preconditioned GMRES methods for solving linear system of equations arising from the discretization of parabolic equations. With these methods, a block diagonal linear system instead of a block lower bidiagonal linear system will be solved in each space–time subdomain. It shows that this technique can reduce the cost of the computation and memory overhead. We develop an optimal convergence theory which shows that convergence rate is bounded independently of the spatial mesh sizes, the time step size, the number of subdomains, the number of levels and the window size. Some numerical results are reported to confirm the theory in terms of optimality and scalability. Moreover, numerical comparisons with space–time additive Schwarz algorithm are also given to show the actual competitiveness our proposed space–time methods.