Time-parallel iterative solvers for parabolic evolution equations

Martin Neumüller (1) , Iain Smears (2)

1 JKU - University of Linz - Johannes Kepler Universität Linz
2 UCL - University College of London [London]

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 analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also 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.

Mots clés: Parabolic partial differential equations; parallel algorithms; preconditioners; convergence; parallel complexity

Domaines: Analyse numérique [math.NA]

Martin Neumüller, Iain Smears. Time-parallel iterative solvers for parabolic evolution equations. SIAM Journal on Scientific Computing, 2019, 41 (1), pp.C28-C51. ⟨10.1137/18M1172466⟩. ⟨hal-02059135⟩