Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations

TL;DR

This paper demonstrates the effectiveness of a local Vanka smoother within a geometric multigrid preconditioner for efficiently solving linear systems arising from space-time finite element discretizations of the Navier-Stokes equations, especially in 2D and 3D flow around a cylinder.

Contribution

It introduces and numerically validates a Vanka smoother-based multigrid preconditioner for high-order space-time finite element methods applied to Navier-Stokes equations.

Findings

Preconditioned GMRES with Vanka smoother shows high efficiency.

Solver robustness is maintained across different polynomial orders in time.

Numerical experiments confirm solver effectiveness for 2D and 3D benchmark flows.

Abstract

Numerical simulation of incompressible viscous flow, in particular in three space dimensions, continues to remain a challenging task. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Linearizing the resulting algebraic problems by Newton's method yields linear systems with block matrices built of $(k+1)\times (k+1)$ saddle point systems, where $k$ denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a $V$-cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two- and three-dimensional benchmark problem of flow around a cylinder. Here, the robustness of the solver with respect to the piecewise polynomial order $k$ in time is analyzed and proved numerically.