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.

我们引入了一个用于求解三角形网格表面上的一类抛物型偏微分方程的框架，包括 Hamilton-Jacobi 方程和 Fokker-Planck 方程。此类中的偏微分方程通常具有非线性或刚性项，无法使用弯曲三角形网格上的标准方法求解。为了应对这一挑战，我们利用分裂积分器与凸优化步骤相结合来求解这些偏微分方程。我们的机器可用于计算几何域上最佳传输距离的熵近似，克服了最先进方法的数值限制。此外，我们还证明了我们的方法在几何处理中的扩散和前传播任务中出现的许多线性和非线性偏微分方程上的多功能性。