A geometric multigrid method for space-time finite element discretizations of the Navier-Stokes equations and its application to 3d flow simulation

TL;DR

This paper introduces a parallelized geometric multigrid method with a Vanka smoother for higher order space-time finite element discretizations of the Navier-Stokes equations, demonstrating efficiency in 2D and 3D flow simulations.

Contribution

It develops a novel GMG solver with a cell-based Vanka smoother tailored for space-time finite element methods applied to Navier-Stokes equations.

Findings

Effective in 2D and 3D flow benchmarks

Demonstrates scalability and efficiency

Compatible with deal.II finite element library

Abstract

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier--Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for GMRES iterations. Its performance properties are demonstrated for 2d and 3d benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the \emph{deal.II} finite element library. The concepts are flexible and can be transferred to similar software platforms.