"Fast Eigensolver for Computing planets' Internal Normal Modes"
The study of Earth's free oscillations is important for understanding Earth's dynamic response to earthquakes and post-seismic relaxation. Indeed, the spectrum of a planet contains rich information about its large-scale structure and provides constraints on heterogeneity in geochemical composition, temperature, and anisotropy. The relatively low eigenfrequencies can be well detected and thus play a vital role in this context.
We develop a novel parallel computational method to compute the planet’s internal normal modes, on an unstructured tetrahedral mesh. We apply the continuous Galerkin (CG) finite element method to the elastic-gravitational system. In the fluid regions, we use a representation of the displacement for describing the oscillations of the non-seismic modes. Effectively, we introduce a spectral cutoff, separating out the essential spectrum. We utilize a novel Lanczos approach with polynomial filtering for solving this generalized eigenvalue problem. Comparing with the standard shift-invert method, the polynomial filtering technique is an ideal candidate for solving three-dimensional large-scale interior eigenvalue problems since it is designed to reduce the memory cost. The matrix-free scheme allows us to deal with fluid separation and self-gravitation efficiently. In our computational experiments, we compare our results with the radial earth model benchmark, and we visualize the normal modes using vector plots for illustrating the properties of the displacements in different modes.