Rigorous convergence proof of space-time multigrid with coarsening in space

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