A15. Bird, P., and J. Baumgardner (1983) 3-D Finite-element modeling of the Earth's free oscillations (abstract), __Eos Trans. AGU__, **64**, 754.

A fast multigrid algorithm is used to solve the generalized eigenvalue problem A u_{i} = l _{i} D u_{i} for the eigenvalues and eigenfunctions of a laterally heterogeneous Earth. The solution technique utilizes properties of a pseudo-inverse A_{n} ^{-1} of the shifted operator A_{n} = A - n D to isolate a cluster of eigenfunctions in the neighborhood of n . The pseudo-inverse is realized by means of a multigrid approximate inverse algorithm developed by P. Fredrikson of Los Alamos National Laboratory. The algorithm requires but O(n) machine operations, where n is the number of grid points. The spherical discretization consists of successive dyadic refinements of the mesh produced by projecting the regular icosahedron onto the sphere. Finite-element basis functions are defined in terms of spherical barycentric coordinates. Computation times on a Cray-1 on the order of minutes are obtained for a grid with 43560 nodes (2562 nodes on each of 17 layers) for modes of Legendre degree 10 or less. Accuracy of the technique is demonstrated by calculating torroidal modes of a uniform spherical shell. Results will be presented for the splitting of the low degree fundamental torroidal modes by the heterogeneity of Jordan's tectosphere model for an Earth otherwise described by PREM.