Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations

Standard Krylov subspace solvers for self-adjoint problems have rigorous convergence bounds based solely on eigenvalues. However, for non-self-adjoint problems, eigenvalues do not determine behavior even for widely used iterative methods. In this paper, we discuss time-dependent PDE problems, which are always non-self-adjoint. We propose a block circulant preconditioner for the all-at-once evolutionary PDE system which has block Toeplitz structure. Through reordering of variables to obtain a symmetric system, we are able to rigorously establish convergence bounds for MINRES which guarantee a number of iterations independent of the number of time-steps for the all-at-once system. If the spatial differential operators are simultaneously diagonalizable, we are able to quickly apply the preconditioner through use of a sine transform, and for those that are not, we are able to use an algebraic multigrid process to provide a good approximation. Results are presented for solution to both the heat and convection diffusion equations.