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

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