Abstract
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.
Similar content being viewed by others
References
Adams, M., Brezina, M., Hu, J., Tuminaro, R.: Parallel multigrid smoothing: polynomial versus Gauss–Seidel. J. Comput. Physics 188, 593–610 (2003)
Baker, A. H., Falgout, R. D., Kolev, T. V., Yang, U. M.: Multigrid smoothers for ultraparallel computing. SIAM J. Sci. Comput. 33, 2864–2887 (2011)
Dobrev, V. A., Kolev, T., Petersson, N. A., Schroder, J. B.: Two-level convergence theory for multigrid reduction in time (MGRIT). SIAM J. Sci. Comput. 39, S501–S527 (2017)
Falgout, R. D., Friedhoff, S., Kolev, T. V., MacLachlan, S. P., Schroder, J. B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36, C635–C661 (2014)
Falgout, R. D., Friedhoff, S., Kolev, T. V., MacLachlan, S. P., Schroder, J. B., Vandewalle, S.: Multigrid methods with space–time concurrency. Comput. Vis. Sci. 18, 123–143 (2017)
Franco, S. R., Gaspar, F. J., Villela Pinto, M. A., Rodrigo, C.: Multigrid method based on a space-time approach with standard coarsening for parabolic problems. Appl. Math. Comput. 317, 25–34 (2018)
Friedhoff, S., MacLachlan, S.: A generalized predictive analysis tool for multigrid methods. Numer. Linear Algebra Appl. 22, 618–647 (2015)
Friedhoff, S., MacLachlan, S., Börgers, C.: Local fourier analysis of space-time relaxation and multigrid schemes. SIAM J. Sci. Comput. 35, S250–S276 (2013)
Gander, M. J., Neumüller, M.: Analysis of a new space-time parallel multigrid algorithm for parabolic problems, SIAM. J. Sci. Comput. 38, A2173–A2208 (2016)
Hackbusch, W.: Parabolic multi-grid methods. In: Proceedings of the Sixth Int’l. Symposium on Computing Methods in Applied Sciences and Engineering, VI, NLD, pp 189–197. North-Holland (1985)
Horton, G.: The time-parallel multigrid method. Comm. Appl. Num. Methods 8, 585–595 (1992)
Horton, G., Vandewalle, S.: A space-time multigrid method for parabolic partial differential equations. SIAM J. Sci. Comput. 16, 848–864 (1995)
Janssen, J., Vandewalle, S.: Multigrid waveform relaxation on spatial finite element meshes: the discrete-time case. SIAM J. Sci. Comput. 17, 133–155 (1996)
Kaveh, A., Rahami, H.: Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures. Acta Mech 217, 51–62 (2011)
Lubich, C., Ostermann, A.: Multi-grid dynamic iteration for parabolic equations. BIT 27, 216–234 (1987)
Southworth, B. S.: Necessary conditions and tight two-level convergence bounds for parareal and multigrid reduction in time. SIAM J. Matrix Anal. Appl. 40, 564–608 (2019)
Stüben, K., Trottenberg, K.U.: Multigrid methods: Fundamental algorithms, model problem analysis and applications. In: Hackbusch, W., Trottenberg, U. (eds.) Lectures Notes in Mathematics No. 960, Berlin Heidelberg New York, pp 1–176. Springer (1982)
Trottenberg, U., Oosterlee, C. W., Schüller, A.: Multigrid. Academic Press, London (2001)
Vandewalle, S., Horton, G.: Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods. Computing 54, 317–330 (1995)
Vandewalle, S., Van de Velde, E.: Space-time concurrent multigrid waveform relaxation. Ann. Numer. Math. 1, 347–360 (1994)
Acknowledgements
We thank an anonymous referee for careful reading and useful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Yvan Notay is Research Director of the Fonds de la Recherche Scientifique – FNRS.
Rights and permissions
About this article
Cite this article
Notay, Y. Rigorous convergence proof of space-time multigrid with coarsening in space. Numer Algor 89, 675–699 (2022). https://doi.org/10.1007/s11075-021-01129-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-021-01129-2