Wave Computation on the Hyperbolic Double Doughnut

TL;DR

This paper presents a finite element method for computing wave propagation on a genus 2 hyperbolic surface, analyzing eigenvalues, decay, and ergodic properties of the wave dynamics.

Contribution

It introduces a variational finite element approach to simulate waves on hyperbolic surfaces with complex topology, including implementation of boundary conditions for Fuchsian groups.

Findings

First eigenvalues of the Laplace-Beltrami operator computed

Exponential decay observed with localized damping

Ergodic behavior of geodesic flow confirmed

Abstract

We compute the waves propagating on the compact surface of constant negative curvature and genus 2. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. A spectral analysis of the wave allows to compute the first eigenvalues of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping and the ergodicity of the geodesic flow.