We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

翻译：我们提出一个框架，用于在三角形网格曲面上求解一类抛物型偏微分方程，包括Hamilton-Jacobi方程和Fokker-Planck方程。这类偏微分方程通常包含非线性或刚性项，无法用标准方法在曲面三角形网格上求解。为应对这一挑战，我们采用分裂积分器结合凸优化步骤来求解这些偏微分方程。我们的方法可用于计算几何域上最优传输距离的熵近似，克服了现有技术方法的数值局限性。此外，我们还在几何处理中扩散和前沿传播任务所涉及的多个线性和非线性偏微分方程上展示了该方法的通用性。