Space-time waveform relaxation multigrid for Navier-Stokes

TL;DR

The paper addresses the challenge of efficiently solving space-time discretized incompressible Navier–Stokes equations by developing a waveform relaxation multigrid (WRMG) framework. It extends efficient spatial relaxation schemes to space-time finite-element discretizations, forming a monolithic Newton–Krylov–multigrid solver that operates on tensor-product space-time grids and uses patch-based relaxation (vertex-star for heat; Vanka+star for Navier–Stokes). The approach is validated on 2D problems (heat, Chorin NS, lid-driven cavity), showing scalability with discretization order, mesh size, and Reynolds number, and offering memory advantages over direct solvers. The results highlight the method’s potential for space-time parallelism, while noting current 2D limitations and future work toward true 3D and deeper space-time parallelism via cyclic reduction strategies.

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 [37] to a waveform relaxation method, and demonstrate the efficiency of the resulting monolithic Newton-Krylov-multigrid solver. Numerical results demonstrate the scalability of the solver for varying discretization order and physical parameters.

Figures (2)

• Figure 1: At left, a sample spatial vertex-star patch used for relaxation for the $\mathbb{P}_{2}$ elements for the heat equation. At right, a sample Vanka+star patch for the $(\mathbb{P}_{3})^2$-$\mathbb{P}_{2}$ discretization of the Navier-Stokes equations.
• Figure 2: Time-to-solution (in seconds) for temporal degree $0$ and spatial degree $1$ for a simulation of the lid-driven cavity Naiver-Stokes test case with $R=10$ for differing numbers of MPI ranks. Here, we compare solving the global space-time system using WRMG as a preconditioner for GMRES against localizing the system to a single timestep and solving sequentially over the timesteps.

Theorems & Definitions (9)

• Definition 2.1: Discontinuous Galerkin notation
• Definition 2.2: Temporal discretisation
• Remark 2.3: Efficient implementation in time
• Remark 2.4: Conforming temporal discretisation
• Definition 2.5: Heat equation
• Definition 2.6: Navier-Stokes equations
• Definition 2.7: Chorin test problem Chorin1968
• Remark 2.8: Dependence on Reynolds number
• Definition 2.9: Lid-driven cavity Bozeman1973