We present a space-time multigrid method based on tensor-product space-time finite element discretizations. The method is facilitated by the matrix-free capabilities of the {\ttfamily deal.II} library. It addresses both high-order continuous and discontinuous variational time discretizations with spatial finite element discretizations. The effectiveness of multigrid methods in large-scale stationary problems is well established. However, their application in the space-time context poses significant challenges, mainly due to the construction of suitable smoothers. To address these challenges, we develop a space-time cell-wise additive Schwarz smoother and demonstrate its effectiveness on the heat and acoustic wave equations. The matrix-free framework of the {\ttfamily deal.II} library supports various multigrid strategies, including $h$-, $p$-, and $hp$-refinement across spatial and temporal dimensions. Extensive empirical evidence, provided through scaling and convergence tests on high-performance computing platforms, demonstrate high performance on perturbed meshes and problems with heterogeneous and discontinuous coefficients. Throughputs of over a billion degrees of freedom per second are achieved on problems with more than a trillion global degrees of freedom. The results prove that the space-time multigrid method can effectively solve complex problems in high-fidelity simulations and show great potential for use in coupled problems.

翻译：我们提出了一种基于张量积时空有限元离散化的时空多重网格方法。该方法利用了{\ttfamily deal.II}库的无矩阵计算能力。它适用于高阶连续及不连续变分时间离散化与空间有限元离散化的结合。多重网格方法在大规模稳态问题中的有效性已得到充分证实。然而，其在时空背景下的应用面临重大挑战，主要源于合适光滑算子的构建。为应对这些挑战，我们开发了一种时空单元加性施瓦茨光滑算子，并在热传导方程和声波方程上验证了其有效性。{\ttfamily deal.II}库的无矩阵框架支持多种多重网格策略，包括跨空间和时间维度的$h$-, $p$-, 和$hp$-细化。通过在高性能计算平台上进行的扩展性和收敛性测试提供的大量实验证据表明，该方法在扰动网格以及具有异质和不连续系数的问题上均表现出高性能。在具有超过万亿全局自由度的问题上，实现了每秒超过十亿自由度的吞吐量。结果证明，时空多重网格方法能够有效解决高保真仿真中的复杂问题，并在耦合问题中展现出巨大应用潜力。