超几何空间上的离散拉普拉斯算子——比较研究
Discrete Laplacians on the hyperbolic space -- a comparative study
AI总结：
本文研究了在二维超几何空间上构造离散拉普拉斯-贝尔特里算子的方法，通过两种有限差分算子的比较，证明了其在连续热方程中的稳定性和收敛性，并展示了其在理论和计算上的优势。
摘要：
本文关注于构造可用于求解偏微分方程的黎曼流形上离散拉普拉斯-贝尔特里算子的构造。由于现有构造往往缺乏严格的收敛保证或计算成本高，本文聚焦于设计计算可行且有收敛结果的算子。以二维超几何空间$\mathbb{H}^2$为起点，开发了两种针对该常负曲率空间的离散有限差分算子，作为连续拉普拉斯-贝尔特里算子在$\mathrm{L}^2$框架下的近似。证明了与这两种算子相关的离散热方程在稳定性上具有收敛性，并趋近于连续热-贝尔特里柯西问题在$\mathbb{H}^2$上的解。此外，通过受离散函数不等式尖锐分析启发的技术，证明了两种离散拉普拉斯算子对应的离散热方程解具有指数衰减，其速率与$\mathbb{H}^2$上的皮卡尔不等式诱导的衰减相同。最终，本文展示了专门设计用于超几何空间几何的离散拉普拉斯算子在理论和计算上的优势，并可有效推广到三维超几何空间。
英文摘要：
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 Poincaré 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.
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