Title: Discrete Laplacians on the hyperbolic space -- a compared study Show Chinese title

Title: 双曲空间上的离散拉普拉斯算子 -- 一项比较研究

Abstract: The main motivation behind this paper stems from a notable gap in the existing literature: the absence of a discrete counterpart to the Laplace-Beltrami operator on Riemannian manifolds, which can be effectively used to solve PDEs. We consider that the natural approach to pioneer this field is first to explore one of the simplest non-trivial (i.e., non-Euclidean) scenarios, specifically focusing on the $2$-dimensional hyperbolic space $\mathbb{H}^2$. We present 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. Show Chinese abstract

Abstract: 本文的主要动机源于现有文献中一个显著的空白：在黎曼流形上缺乏拉普拉斯-贝尔特拉米算子的离散对应物，而这种对应物可以有效地用于求解偏微分方程。我们认为，开拓这一领域的自然方法是首先探索一个最简单的非平凡（即非欧几里得）场景，具体来说是专注于$2$维双曲空间$\mathbb{H}^2$。我们提出了两种针对这种常负曲率空间的离散有限差分算子，两者都在$\mathrm{L}^2$框架内作为（连续）拉普拉斯-贝尔特拉米算子的近似。我们证明了与上述两种算子相关的离散热方程表现出稳定性，并且收敛到$\mathbb{H}^2$上的连续热-贝尔特拉米柯西问题。此外，利用离散函数不等式尖锐分析的技巧，我们证明了对应于两种离散拉普拉斯算子的离散热方程的解在渐近意义上呈现出与$\mathbb{H}^2$上庞加莱不等式所诱导的指数衰减相等的衰减。最终，我们展示了专为双曲空间几何设计的离散拉普拉斯算子能够提供更精确的近似，并在理论和计算方面都具有优势。此外，这种离散算子可以有效地推广到三维双曲空间。