Wave computation on the Poincaré dodecahedral space
This work addresses the problem of simulating wave propagation on a specific cosmological model for researchers studying the topology of the universe.
The authors computed wave propagation on the Poincaré dodecahedral space, a compact 3-manifold with constant positive curvature and non-trivial topology, which is a plausible model for a multi-connected universe. The computation was validated through spectral analysis by computing many eigenvalues of the Laplace-Beltrami operator.
We compute the waves propagating on a compact 3-manifold of constant positive curvature with a non trivial topology: the Poincaré dodecahedral space that is a plausible model of multi-connected universe. We transform the Cauchy problem to a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We adopt a variational approach using a space of finite elements that is invariant under the action of the binary icosahedral group. The computation of the transient waves is validated with their spectral analysis by computing a lot of eigenvalues of the Laplace-Beltrami operator.