The computation of seismic normal modes with rotation as a quadratic eigenvalue problem. (arXiv:1912.00114v1 [physics.comp-ph])

<a href="http://arxiv.org/find/physics/1/au:+Shi_J/0/1/0/all/0/1">Jia Shi</a>, <a href="http://arxiv.org/find/physics/1/au:+Li_R/0/1/0/all/0/1">Ruipeng Li</a>, <a href="http://arxiv.org/find/physics/1/au:+Xi_Y/0/1/0/all/0/1">Yuanzhe Xi</a>, <a href="http://arxiv.org/find/physics/1/au:+Saad_Y/0/1/0/all/0/1">Yousef Saad</a>, <a href="http://arxiv.org/find/physics/1/au:+Hoop_M/0/1/0/all/0/1">Maarten V. de Hoop</a>

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 fluid 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 from a non-rotating 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 to be 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 are

used to compute the point spectra of eigenfrequencies in several Mars models

studying the effects of heterogeneity on a large range of scales.

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 fluid 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 from a non-rotating 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 to be 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 are

used to compute the point spectra of eigenfrequencies in several Mars models

studying the effects of heterogeneity on a large range of scales.

http://arxiv.org/icons/sfx.gif