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 generalized minimal residual 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 deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.

我们提出了一种基于单元格 Vanka 平滑器的并行几何多重网格 (GMG) 方法，用于不可压缩 Navier-Stokes 方程的高阶时空有限元方法 (STFEM)。STFEM 作为时间推进方案实施。GMG 求解器用作广义最小残差迭代的预处理器。其性能属性已针对圆柱体周围流动的 2D 和 3D 基准进行了演示。GMG 方法的关键要素是在各个子区间的所有时间自由度上构建局部 Vanka 平滑器及其有效应用。为此，生成了存储所有层次级别的雅可比行列式的预计算单元逆并且仅需要合理数量的内存开销的数据结构。GMG 方法是为deal.II有限元库。这些概念是灵活的，可以转移到类似的软件平台。