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.

翻译：我们以基于单元格的Vanka平滑器为基础,为无法压缩的导航-斯托克方程式提供了一种平行的几何多格多格(GMG)方法。 STFEM 是一个时间进化方案。 GMG求解器用作GMRES迭代的前提条件。 它的性能特征为圆柱圆圆周围的2d和3d流动基准。 GMG 方法的关键成分是,在相应次间隙及其有效应用期间,在各种自由度上建造本地的Vanka光滑器。 为此,所有等级层都储存了预合成单元格反向Jacobian的数据结构,只要求生成合理的内存管理量。 GMG方法是为\emph{deal.II} 有限元素库构建的。 概念是灵活的,可以转移到类似的软件平台 。