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


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.


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We would like to thank Joost van Zwieten, co-developer of the open source project nutils Footnote 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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Correspondence to M. Baumann.

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Baumann, M., Astudillo, R., Qiu, Y. et al. An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies. Comput Geosci 22, 43–61 (2018).

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