AI 中文总结
研究二维里奇流有限元方法的收敛性，通过将其表述为解驱动的度量演化，利用高斯曲率驱动，经有限元离散，采用特定矩阵向量公式证明收敛，还能保留重要几何结构，数值实验验证了方法有效性。
AI 中文摘要
证明了二维里奇流有限元离散化的收敛性。此方法将二维曲面上的里奇流表述为解驱动的度量演化，由高斯曲率驱动度量演化。高斯曲率满足抛物方程且依赖于度量，增强了问题的抛物结构。通过有限元方法离散解驱动的度量演化公式，并采用文献中最初用于研究外在曲率流中解驱动表面演化的矩阵向量公式证明了有限元近似的收敛性。该方法除收敛性外，还在离散层面保留了里奇流的重要几何结构，如面积守恒和高斯 - 博内定理。给出了大量数值实验以证明该方法的收敛性及里奇流模拟。
英文摘要
The convergence of a finite element discretization for the two-dimensional Ricci flow is proved. In this method, the Ricci flow on a two-dimensional surface is formulated into solution-driven metric evolution, with the metric evolution driven by the Gauss curvature. The Gauss curvature satisfies a parabolic equation that in turn depends on the metric, thereby enhancing the parabolic structure of the problem. The solution-driven metric evolution formulation is discretized by the finite element method, and the convergence of finite element approximations is proved by adapting the matrix-vector formulation developed in the literature initially for studying solution-driven surface evolution in extrinsic curvature flow. In addition to its convergence, the proposed method also preserves important geometric structures of the Ricci flow at the discrete level, such as area conservation and the Gauss-Bonnet theorem. Extensive numerical experiments are presented to demonstrate the convergence of the proposed method as well as the simulation of Ricci flow.
Comments31 pages, 2 figures
