A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

Original languageEnglish (US)
Article number67
JournalJournal of Scientific Computing
Issue number2
StatePublished - May 2022
Externally publishedYes

Bibliographical note

Funding Information:
We would like to thank Bernard Valette and anonymous referees for their thoughtful comments. J.S. would like to thank Petroleum Geo-Services for using their supercomputer Abel, Danny Sorensen, Ruichao Ye, Harry Matchette-Downes, Anton Ermakov, and Burkhard Militzer for helpful discussions.

Funding Information:
This research was supported by the Simons Foundation under the MATH+X program, the National Science Foundation Grant DMS-1815143, the members of the Geo-Mathematical Imaging Group at Rice University, and XSEDE research allocation TG-EAR170019. The work by R.L. was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL-JRNL-780818). Y.X. and Y.S. were supported by NSF-1812695.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


  • Earth and planetary sciences
  • Eigensolver
  • Normal modes
  • Polynomial filtering


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