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

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.