Space-Time Waveform Relaxation Multigrid for Navier–Stokes

Abstract. Space-time finite-element discretizations are well-developed in many areas of science and engineering, but much work remains within the development of specialized solvers for the resulting linear and nonlinear systems. In this work, we consider the all-at-once solution of the discretized Navier–Stokes equations over a space-time domain using waveform relaxation multigrid methods. In particular, we show how to extend the efficient spatial multigrid relaxation methods from Rafiei and MacLachlan [preprint, arXiv:2502.01130] to a waveform relaxation method, and provide a proof-of-concept of the algorithmic efficiency of the resulting monolithic Newton–Krylov-multigrid solver. Numerical results demonstrate the scalability of the solver for varying discretization order and physical parameters. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/JamesJackaman/RelaxNavierStokes/tree/main and in the supplementary materials ( RelaxNavierStokes-main.zip [37.1KB]). [Formula: see text]