A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

Abstract

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. Certain PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on triangle mesh surfaces. 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 tasks in geometry processing.