Discrete Laplacians on the hyperbolic space -- a comparative study

TL;DR

This paper develops and compares two discrete Laplacian operators on hyperbolic space, demonstrating their stability, convergence, and improved approximation qualities for solving heat equations numerically.

Contribution

It introduces two novel finite-difference operators tailored to hyperbolic geometry, with proven convergence and stability, and highlights a more precise operator for hyperbolic spaces.

Findings

Both operators ensure stability and convergence of the discrete heat equation.

Solutions decay exponentially, matching Poincaré inequality predictions.

A geometry-specific discrete Laplacian provides better approximation and computational advantages.

Abstract

This paper is concerned with the construction of discrete counterparts of the Laplace-Beltrami operator on Riemannian manifolds that can be effectively used in the numerical solution of partial differential equations. Since existing constructions often lack rigorous convergence guarantees or imply a significant computational effort, we focus on designing operators that are both computationally feasible and supported by convergence results. We consider as a starting point the two-dimensional hyperbolic space $\mathbb{H}^2$, one of the simplest non-Euclidean settings, and develop two variants of discrete finite-difference operator tailored to this constant negatively curved space, both serving as approximations to the (continuous) Laplace-Beltrami operator within the $\mathrm{L}^2$ framework. We prove that the discrete heat equation associated with both operators mentioned above exhibits stability and converges towards the continuous heat-Beltrami Cauchy problem on $\mathbb{H}^2$. Moreover, using techniques inspired from the sharp analysis of discrete functional inequalities, we prove that the solutions of the discrete heat equations corresponding to both variants of discrete Laplacian exhibit an exponential decay asymptotically equal to the one induced by the Poincar\'e inequality on $\mathbb{H}^2$. Eventually, we illustrate that a discrete Laplacian specifically designed for the geometry of the hyperbolic space yields a more precise approximation and offers advantages from both theoretical and computational perspectives. Furthermore, this discrete operator can be effectively generalized to the three-dimensional hyperbolic space.