Computational Geosciences

, Volume 22, Issue 1, pp 43–61 | Cite as

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

  • M. BaumannEmail author
  • R. Astudillo
  • Y. Qiu
  • E. Y. M. Ang
  • M. B. van Gijzen
  • R.-É. Plessix
Open Access
Original Paper


In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.


Time-harmonic elastic wave equation Multiple frequencies Induced dimension reduction (IDR) method Preconditioned matrix equations Multilevel sequentially semiseparable (MSSS) matrices 



We would like to thank Joost van Zwieten, co-developer of the open source project nutils 1 for helpful discussions concerning the finite element discretization described in Section 2.2. Shell Global Solutions International B.V. is gratefully acknowledged for financial support of the first author.


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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Delft University of TechnologyDelftThe Netherlands
  2. 2.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  3. 3.Nanyang Technological UniversitySingaporeSingapore
  4. 4.Shell Global Solutions International B.V.The HagueThe Netherlands

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