Space-time waveform relaxation multigrid for Navier-Stokes

TL;DR

This paper develops a space-time waveform relaxation multigrid method for efficiently solving the Navier-Stokes equations all-at-once, demonstrating scalability and effectiveness through numerical experiments.

Contribution

It extends spatial multigrid relaxation techniques to a waveform relaxation framework for Navier-Stokes, creating a scalable monolithic solver.

Findings

The solver is scalable with respect to discretization order.

Numerical results confirm the efficiency of the method.

The approach effectively handles varying physical parameters.

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.