The computation of seismic normal modes with rotation as a quadratic eigenvalue problem [preprint]

Preprint date

January 28, 2020

Authors

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad (professor) and Maarten V. de Hoop

Abstract

A new approach is presented to compute the seismic normal modes of a fully heterogeneous, rotating planet. Special care is taken to separate out the essential spectrum in the presence of a liquid outer core. 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 (FMM). To solve the resulting quadratic eigenvalue problem (QEP), the approach utilizes extended Lanczos vectors forming a subspace computed in a nonrotating planet – with the shape of boundaries of a rotating planet and accounting for the centrifugal potential – to reduce the dimension of the original problem significantly. The subspace is guaranteed being contained in the space of functions to which the seismic normal modes belong. The reduced system can further be solved with a standard eigensolver. The computational accuracy is illustrated using all the modes with relative small meshes, and also tested against standard perturbation calculations relative to a standard Earth model. The algorithm and code is used to compute the point spectra of eigenfrequencies in several Mars models studying the effects of heterogeneity on a large range of scales.

Link to full paper

The computation of seismic normal modes with rotation as a quadratic eigenvalue problem

Keywords

eigenvalue problems