# A non-perturbative approach to computing seismic normal modes in rotating planets

## TL;DR

This paper introduces a non-perturbative, high-performance computational method for accurately determining seismic normal modes in rotating planets, incorporating complex physics and validated against known models.

## Contribution

It presents a novel finite-element based approach with advanced eigensolvers for seismic mode computation in rotating planets, including fluid and solid regions, validated against analytical and numerical benchmarks.

## Key findings

- Validated methods against analytical solutions for non-rotating cases.
- Successfully computed seismic modes in Earth and Mars models.
- Analyzed effects of heterogeneity on seismic eigenfrequencies.

## Abstract

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.