Space-time multigrid refers to the use of multigrid methods to solve discretized partial differential equations considering at once multiple time steps. A new theoretical analysis is developed for the case where one uses coarsening in space only. It proves bounds on the 2-norm of the iteration matrix that connect it to the norm of the iteration matrix when using the same multigrid method to solve the corresponding stationary problem. When using properly defined wavefront type smoothers, the bound is uniform with respect to the mesh size, the time step size, and the number of time steps, and addresses both the two-grid case and the W-cycle. On the other hand, for time-parallel smoothers, the results clearly show the condition to be satisfied by the time step size to have similar performance as with wavefront type smoothers. The analysis also leads to the definition of an effective smoothing factor that allows one to quickly check the potentialities of a given smoothing scheme. The accuracy of the theoretical estimates is illustrated on a numerical example, highlighting the relevance of the effective smoothing factor and the usefulness in following the provided guidelines to have robustness with respect to the time step size.

时空多重网格是指使用多重网格方法求解同时考虑多个时间步长的离散偏微分方程。针对仅在空间中使用粗化的情况开发了一种新的理论分析。当使用相同的多重网格方法解决相应的平稳问题时，它证明了迭代矩阵的2-范数与迭代矩阵的范数相连的界限。当使用正确定义的波前类型平滑器时，边界在网格大小、时间步长和时间步数方面是一致的，并且解决了双网格情况和 W 循环。另一方面，对于时间平行平滑器，结果清楚地表明时间步长满足的条件与波前型平滑器具有相似的性能。该分析还导致了一个有效平滑因子的定义，它允许人们快速检查给定平滑方案的潜力。数值示例说明了理论估计的准确性，突出了有效平滑因子的相关性以及遵循提供的指南以在时间步长方面具有鲁棒性的有用性。