Towards an intrinsic geometrical description of elastic dynamics.

The spectral element method (SEM) has been one of the main numerical tools for simulating seismic waves. Unfortunately, there is not an intrinsic way for extending the high accuracy discretization of space performed by SEM, to the time variable. The idea of using geometric integration for long term studies has been proposed for several authors, but no consistent (intrinsically defined) theory has been developed for elastic dynamics. In this talk we recall some facts about SEM and some basic ideas of geometrical integration. We will expose briefly new results concerning some geometrical integratos called "symplectic integrators" and the perspective for finding an intrinsic numerical method for elastic dynamics.