Wave computation on the Poincar\'e dodecahedral space

TL;DR

This paper models wave propagation on the Poincaré dodecahedral space, a candidate shape for the universe, using finite element methods and spectral analysis to validate the results.

Contribution

It introduces a variational finite element approach on a fundamental domain for wave computation on a complex 3-manifold with positive curvature.

Findings

Validated wave computations with spectral eigenvalues

Computed eigenvalues of the Laplace-Beltrami operator

Demonstrated the feasibility of numerical modeling on complex topologies

Abstract

We compute the waves propagating on a compact 3-manifold of constant positive curvature with a non trivial topology: the Poincar\'e 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.