2024/09/02 by Mihai Bucataru, Bucataru, Mihai, Dragoş Manea +1
Computer Science · Mathematics · #35K08 #58J35 #58J60 #65M06 #65M15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2409.01211
openalex publication_date 2024/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 ℍ2, 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 L2 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 ℍ2. 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 Poincaré inequality on ℍ2. 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.