Time-Parallel Iterative Solvers for Parabolic Evolution Equations

Abstract

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyze an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is nonintrusive and easy to implement in practice. It also features the key theoretical advantage of robust spectral bounds, which lead to convergence rates that are independent of the number of time-steps, final time, and spatial mesh size. Finally, it has a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties.